OPTIMAL INVESTMENT AND CONSUMPTION PORTFOLIO...

OPTIMAL INVESTMENT AND CONSUMPTION PORTFOLIO CHOICE PROBLEM FORASSETS MODELED BY LEVY PROCESSES

By

RYAN G. SANKARPERSAD

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2011

c⃝ 2011 Ryan G. Sankarpersad

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ACKNOWLEDGMENTS

I would like to thank my dissertation advisor Dr Liqang Yan for his help throughout

the research process, your guidance was crucial to my growth as a mathematician.

Also Dr Yan’s courses in stochastic calculus and mathematical finance helped

develop my interest in the field of finance. I would also like to individually thank all

members of my dissertation committee. Dr Michael Jury’s year long class in Measure

Theory was fundamental to my understanding of more advanced concepts required

for this dissertation. Dr Murali Rao’s course in Probability theory helped solidify my

understanding of measure theory and was a great introduction to the theory of random

variables. Dr Stan Uryasev’s courses on fixed income derivatives, portfolio theory and

risk management techniques introduced much of the financial theory that I have used

in this dissertation and throughout the job application process. Dr Joseph Glover’s

poignant questions during my Oral Exam and insight into the field of Levy Processes

were helpful in guiding my research over the last few years of research. I would also

like to thank Dr Jay Ritter from the Department of Finance for teaching me the classical

theory of corporate finance during his year long course. Last but not least, I would

like to thank my family (Mom, Dad, Rianna and Reshma) for all the love and support

throughout the dissertation process, you were all instrumental to my success.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

CHAPTER

1 GENERAL FRAMEWORK AND PROBLEM SETUP . . . . . . . . . . . . . . . 9

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Stochastic Calculus Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 111.3 Ito Diffusion and it’s Generator . . . . . . . . . . . . . . . . . . . . . . . . 14

2 STATEMENT OF MAIN PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Dynamic Programming Methodology . . . . . . . . . . . . . . . . . . . . . 232.3 Hamilton Jacobi Bellman . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 Verification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 MERTON PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Classic Merton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Infinite Horizon Power Utility . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Infinite Horizon Log Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Finite Horizon Power Utility . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Finite Horizon Log Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4 LEVY PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Why use a Levy Process with Jumps? . . . . . . . . . . . . . . . . . . . . 614.2 Preliminaries of Levy Process . . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Ito-Levy Diffusion and it’s Generator . . . . . . . . . . . . . . . . . . . . . 674.4 Verification Theorem for Levy Process with Jumps . . . . . . . . . . . . . 73

5 OPTIMAL CONTROL PROBLEM IN THE JUMP CASE . . . . . . . . . . . . . 77

5.1 Classic Merton Problem with Jumps . . . . . . . . . . . . . . . . . . . . . 775.2 Jump Case with Log Utility . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3 Finite Horizon Power Utility with Jumps . . . . . . . . . . . . . . . . . . . 885.4 Generalization of Bequest Function in the Jump Case . . . . . . . . . . . 93

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6 NUMERICAL RESULTS AND CONCLUSION . . . . . . . . . . . . . . . . . . . 97

6.1 Jump diffusion and Levy triplets . . . . . . . . . . . . . . . . . . . . . . . . 976.2 Optimal θ∗ for Power Utility . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2.1 Γ(1, 1) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2.2 Compound Poisson Process with Exponential Density . . . . . . . 101

6.3 Explicit Value Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5

LIST OF TABLES

Table page

2-1 Table with comparison of two worlds used to solve optimal control problem . . 26

3-1 Maximum differences between finite horizon and infinite horizon for differentvalues of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6-1 Table for θ∗ as a function of σ, where α = .16, r = .06, γ = .5 . . . . . . . . . . 104

6-2 Table for θ∗ as a function of γ, where α = .16, r = .06, σ = .5 . . . . . . . . . . 105

6

LIST OF FIGURES

Figure page

3-1 Value function for power utility with no jumps with parameters, α = .16, r =.06, δ = 1, γ = .5, σ = .3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3-2 Wealth process for power utility with no jumps with parameters, w = 1,α =.16, r = .06, δ = 1, γ = .5, σ = .3 . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3-3 Value function for log utility with no jumps with parameters, α = .16, r =.06, δ = 1, γ = .5, σ = .3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3-4 Wealth process for Log utility with no jumps with parameters, w = 1,α =.16, r = .06, δ = 1, σ = .3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3-5 Plots of the value function for values of T = 1, 2, 100 compared to the plot ofthe value function for infinite horizon . . . . . . . . . . . . . . . . . . . . . . . . 54

5-1 Plots of the value function for values of T = 1, 2.5, 5, 7.5, 100 compared to theplot of the value function in the infinite horizon case. . . . . . . . . . . . . . . . 92

5-2 Plots of the value function for values of λ = 1, .5, .25, .125, 1e-09 compared tothe plot of the value function for λ = 0 . . . . . . . . . . . . . . . . . . . . . . . 96

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

OPTIMAL INVESTMENT AND CONSUMPTION PORTFOLIO CHOICE PROBLEM FORASSETS MODELED BY LEVY PROCESSES

By

Ryan G. Sankarpersad

May 2011

Chair: Liqang YanMajor: Mathematics

We consider an extension of Merton’s optimal portfolio choice and consumption

problem for a portfolio in which the underlying risky asset is an exponential Levy

process. The investor is able to move money between a risk free asset and a risky asset

and consume from the risk free asset. Given the dynamics of the total wealth of the

portfolio we consider the problem of finding portfolio weights and a consumption process

which optimizes the investors expected utility of consumption over the investment

period. The problem is solved in both the finite and infinite horizon cases for a family

of hyperbolic absolute risk aversion utility functions using the techniques of stochastic

control theory. The general closed form solutions are found for for the case of a power

utility function and then for a more generalized utility. We consider a variety of Levy

processes and make a comparison of the optimal portfolio weights. We find that our

results are consistent with expectations that the greater the inherent uncertainty of

a given process leads to a smaller fraction of wealth invested in the risky asset. In

particular an investor is more careful when the risky asset is a discontinuous Levy

process when compared to the continuous case such as those found in a geometric

Brownian motion model.

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CHAPTER 1GENERAL FRAMEWORK AND PROBLEM SETUP

1.1 Introduction

Given a portfolio of assets the problem of finding optimal weights for each of the

assets has been extensively studied and has many real world applications. The first

major breakthrough in this area was made by Henry Markowitz in his 1952 JF paper

[17] in which he was able to find optimal portfolio weights for each of the assets in the

portfolio in a discrete time setting. Markowitz analysis was a one period model that

aimed to minimize the risk under a constraint on the expected return of the portfolio.

He was able to find a so called efficient frontier which was the intersection of the set

of all admissible portfolios with a portfolio with minimum risk and maximum expected

return. One major computational restriction of this model is that one needs a large

data set to compute the necessary covariances between each of the assets in the

portfolio. A successful attempt to extend Markowitz’s model to a continuous time setting

was made by Robert Merton in his 1972 [18] work for which he received the Nobel

prize. In this work Merton showed that if assets in the portfolio were modeled using

geometric Brownian motion, the portfolio of n assets could be reduced to a portfolio

of only two assets using his so called mutual fund theorem. The mutual fund portfolio

consisting of two assets is made up of a risk free asset and a risky asset. The risk free

asset is typically is a U.S. treasury bill while the risky asset is a linear combination of

the n − 1 remaining assets in the portfolio. This idea mirrors the discrete time idea

of Markowitz’s market portfolio which is made up of a combination of assets. Once

this simplification has been made Merton considers the problem of finding optimal

consumption and portfolio weights for the family of hyperbolic absolute risk aversion

utility functions. Once the set of optimal controls are found Merton then finds the

portfolio weights for each of the n assets in the portfolio so that a comparison can be

made with the optimal Markowitz mean-variance portfolio. This procedure will not be the

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a major focus of our paper and we refer the interested reader to [18] for a discussion

in this direction. There have been many studies which have considered relaxing the

conditions of Merton’s original paper. Davis and Norman [10] 1990 paper relaxes the

condition of no transaction costs and present more generalized results to include convex

transaction cost functions . Fleming and Pang [11] consider the Merton problem with

a Vasieck interest rate model and present results using a subsolution/supersolution

methodology. Papers such as [16] and [27] introduce illiquid assets into the market

and present results to the original Merton problem. Each of these papers presents the

classic Merton problem and it’s solution in some context, a trend that we will continue in

this paper. However we will move in a new direction by considering the Merton problem

with a exponential Levy process so that the price process is allowed to have jumps (in

particular is allowed to be discontinuous).

Although we present some results in the geometric Brownian case (classic Merton

case), the main results of our paper are for price processes with discontinuities such

as those considered in Merton’s 1976 paper [19]. In this paper Merton considers prices

processes which are allowed to have Poisson jumps and derives a closed form solution

for the price of vanilla options. This paper was a major breakthrough in the use of

discontinuous price processes, and although will not be concerned directly with option

pricing formulas we will use the ideas presented in this paper. The main results of our

paper comes from modeling the risky asset in our portfolio by a Levy process with jumps

and finding controls which optimize expected utility of consumption. This work closely

follows research performed by Bernt Oksendal which are summarized in his book

applied stochastic control of jump diffusions [21]. Oksendal solves the Merton problem

in the case when the underlying price process is an exponential Levy process, a solution

which we present for comparison and completeness. Once we have presented the

solution in cases covered under Oksendal’s presentation we proceed to generalize

the utility functions and derive closed form solutions to the corresponding problem. A

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derivation of the closed form solution is provided along with numerical results to verify

the accuracy of the solutions provided.

The formulation and solution of our problem requires the study of stochastic

calculus and other theories which require knowledge of stochastic calculus. A complete

survey of stochastic calculus is not provided and it is assumed the reader is familiar

with the general theory found in such books as [20] and [22]. In particular we assume

the reader has had some introductory exposure to stochastic differential equations and

the associated probability theory required to solve these equations. We begin with a

presentation of the stochastic calculus necessary for our analysis, starting with some

introductory definitions and theorems.

1.2 Stochastic Calculus Preliminaries

This section lays the foundation for much of the paper, many of these results are

found in introductory stochastic calculus books such as [20] and [22]. We assume the

reader is familiar with a probability space (Ω,F ,P) where Ω is the set of all possible

outcomes, F is the σ-algebra, and P is a probability measure. A stochastic process

(Xt)t≥0 is a sequence of random variables defined of the probability space (Ω,F ,P)

taking values in R. In particular for fixed time t the map Xt(ω) is a random variable

for each ω ∈ Ω, while for a fixed realization of the event ω ∈ Ω the map Xt(ω) is a

real valued function for each t ≥ 0. As is standard we will suppress the dependence

on ω throughout the paper and write the random variable as Xt . To study a stochastic

process (Xt)t≥0 defined on (Ω,F ,P) we need a way of encompassing the information

the process Xt has accumulated up time t so that decisions can be made about the

process. Information about random variables is classified using the concept of a

σ-algebra, hence for a stochastic process we will need to consider a sequence of

increasing σ-algebras typically referred to as a filtration.

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Definition 1.1. Given a probability space (Ω,F ,P), a filtration is a family F = (Ft)t≥0 of

σ-algebras such that

Fs ⊂ Ft ∀ 0 ≤ s < t

The four tuple (Ω,F ,F,P) is called a filtered probability space.

A process is a sequence of random variables, in order to work with each of these

random variables we need a way of deciding whether a given events probability can

be computed. To make decisions about a random variable up to time t we need that

the random variable Xt be Ft-measurable. A stochastic process is just a sequence of

random variables so we need the following definition for the measurability of the entire

sequence of random variables.

Definition 1.2. A stochastic process (Xt)t≥0 is called Ft-adapted, if Xt is Ft-measurable

for all t ≥ 0.

An important filtration we will be considering is the filtration generated by a process X , it

is defined by

Definition 1.3. The filtration generated by the process X denoted FX is

FX = σ(Xs ; s ∈ [0, t])

This filtration is the smallest σ-algebra for which all the Xs , 0 ≤ s ≤ t are measurable. In

particular we have that the process X is FX -adapted. To define a stochastic differential

equations we will need to consider processes with non zero quadratic variation which

extend the differential equations from Newtonian Calculus. To include terms with non

zero quadratic variation in our diffusion equations we define a stochastic process of

this type called Brownian motion, we will extend this definition later on to include more

general processes (Levy Process).

Definition 1.4. Let (Ω,F , Ftt≥0,P) be a filtered probability space. An Ft adapted

stochastic process (Bt)t≥0 is called a Brownian motion if the following conditions hold

(i) B0 = 0 a.s

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(ii) for t0 < t1 < · · · < tn the random variables Bt1−Bt0, · · · ,Btn−Btn−1 are independent.(Independent increments)

(iii) Bt+h − Bt is normally distributed for all t and h, in particular Bt+h − Bt has a N(0, t)distribution.

(iv) For every ϵ > 0 and h ≥ 0 (stochastic continuity)

limt→hP(|Bt − Bh| > ϵ) = 0

For much of our work we will be considering the filtration generated by Brownian

motion, in particular we will be working with the filtration FB = σ(Bs ; s ∈ [0, t]). When

considering our stochastic differential equations as a diffusion process the Brownian

motion term contains all sources of randomness. Hence the filtration σ(Bs ; s ∈ [0, t])

will contain all information necessary to make decisions about the random component

of the diffusion process. Once we are able to encapsulate the information of a process

in the form of a σ-algebra we are able to develop the idea of the conditional expectation

given a σ-algebra. This concepts allows us to find a subset of stochastic processes

whose expected future value based on given information is equal to its current value

called a martingale. The mathematical formalities of which are captured in the following

definition.

Definition 1.5. A stochastic process (Xt)t≥0 is called a martingale with respect to the

filtration (Ft)t≥0 if for all t ≥ 0

• E [|Xt |] < ∞ i.e. Xt is integrable

• Xt is Ft-measurable for all t

• for any s < t,E [Xt |Fs ] = Xs

We will be interested in two specific examples of martingales in this paper, these are

Brownian motion and the compensated Levy measure N. The fact that Brownian motion

is a martingale process, may be one of the most widely used and useful properties in

mathematical finance. The martingale nature of the compensated Levy measure allows

it to inherit many of the useful properties that Brownian motion possess, and we will

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use these extensively while proving results for general Levy processes later on. We

would like to study the set of all Ito processes which can be written in terms of these two

martingales so that we may replicate the modern portfolio theory. To do this we must

write down the stochastic diffusion equations which models stock prices of our portfolio,

hence we must give a precise mathematical definition of a diffusion equation.

1.3 Ito Diffusion and it’s Generator

To study a portfolio of assets in a continuous time setting we need to describe

the dynamics of the given assets, this can be done by using the theory of stochastic

differential equations. Each asset will be modeled using a diffusion process which

can be thought of as a particle whose trajectory is influenced by an external source of

randomness. This randomness may come from be attributed to one or many external

sources, for the remainder of the paper we will assume that there will be only one source

of randomness. This means that our theorems will be stated in the one dimensional

case, since a one dimensional Brownian motion will be the driving force of randomness

in our diffusion equations. The multidimensional versions of the theorems we use in this

section can be found in an introductory stochastic calculus books such as [20] or [22].

The source of randomness is modeled by adding a Brownian motion term to a standard

diffusion process, more precisely

Definition 1.6. An Ito diffusion is a stochastic process (Xt)[s,T ] satisfying a stochastic

differential equation of the form

dXt = α(t,Xt) dt + σ(t,Xt) dBt t ∈ [s,T ]; Xs = x ∈ R (1–1)

where Bt is an one-dimensional Brownian motion and α : [s,T ]× R → R,

σ : [s,T ] × R → R satisfy the following global Lipschitz and at most linear growth

conditions for all x , y ∈ R, t ∈ [s,T ]

|α(t, x)− α(t, y)|+ |σ(t, x)− σ(t, y)| ≤ C |x − y |

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|α(t, x)|+ |σ(t, x)| ≤ D(1 + |x |)

Moreover if the solution Xt(ω) is adapted to the filtration σ(Bs ; s ≤ t) then

E s,x[∫ Ts

|Xt |2 dt]< ∞

The unique solution of Equation (1–1) will be denoted by X s,xt for all t ≥ s , if s = 0 we

will use the notation X xt . We note that the drift coefficient α and the diffusion coefficient

σ depend on the time parameter t, with a transformation we can reduce this case to the

time dependent case. The reduction will be performed later on in the paper, hence many

of the definitions and theorems will be stated without explicit time dependence. The

solution X s,xt of Equation (1–1) is referred to as time homogeneous the following reason,

X s,xs+h = x +

∫ s+hs

α(r ,X s,xr ) dr +

∫ s+hs

σ(r ,X s,xr ) dBr (1–2)

Making the change of variables u = r − s we may write Equation (1–2) as

X s,xs+h = x +

∫ h0

α(u + s,X s,xu+s) du +

∫ h0

σ(u + s,X s,xu+s) dBu+s

= x +

∫ h0

α(u + s,X s,xu+s) du +

∫ h0

σ(u + s,X s,xu+s) dBu

where dBu = Bu+s − Bs is a Brownian motion with the same P-distribution as Bu. Since

(Bu)u≥0 and (Bu)u≥0 have the same P-distribution a stochastic differential equation of the

form

dXt = α(t,Xt) dt + σ(t,Xt) dBt ; X0 = x

whose solution X 0,xh can be written as

X 0,xh = x +

∫ h0

α(u,X 0,xu ) du +

∫ h0

σ(u,X 0,xu ) dBu (1–3)

we have that the equations governing both of these Ito processes are essentially the

same from a probability standpoint. In particular, comparing Equations (1–2) and (1–3)

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we see that the latter is just the former equation with s = 0 so that X 0,xh has the same

P-distribution as X s,xs+h i.e. (Xt)t≥0 is time homogeneous. We will henceforth refer to both

stochastic differential equations interchangeably. An important property of Ito diffusions

which will be of great use is the so called Markov property which allows us to use the

present behavior of a diffusion to make decisions about the future without considering

past behavior. In order to state the Strong Markov property we need the following

definition of a random time

Definition 1.7. Let (Ft)t≥0 be a filtration, a function τ : Ω → [0,∞) is called a stopping

time with respect to (Ft)t≥0 if

ω : τ(ω) ≤ t ∈ Ft

A stopping time is a random variable for which the set of all paths(events) ω ∈ Ω with

τ(ω) ≤ t can be decided given the filtration Ft . The definition allows us to consider

only the random times for which we may decided whether or not the time has been

reached given the information up to time t i.e. given Ft . Once we have defined this

random time we may state the Strong Markov property

Theorem 1.1. (Strong Markov property for Ito diffusions) Let (Xt)t≥0 be an Ito diffusion,

τ a stopping time and f : Rn → R be a Borel measurable function, then for ω ∈ Ω, h ≥ 0

E x [f (Xτ+h)|Fτ ](ω) = EXτ (ω)[f (Xh)]

Throughout much of the theory of mathematical finance we assume that the price

process of a given asset is the solution of a stochastic differential equation, hence a

stochastic process. We would like to consider functions on these stochastic process,

so the natural question is to ask whether a function on a stochastic process is itself a

stochastic process. In order to answer this question we need to be able to write down

the differential equation associated with this new process. This means that the classical

theory of Newtonian calculus must be extended to include stochastic terms. This is done

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by Ito’s formula which is a generalization of the chain rule from Newtonian calculus, we

will extend to a more general version later in the paper.

Theorem 1.2. (1- dimensional Ito formula) Given an Ito process Xt of the form

dXt = α(t,Xt) dt + σ(t,Xt) dBt

where Bt ∈ R is a Brownian motion. If g(t, x) ∈ C 1,2([0,∞),R) then

Yt = g(t,Xt)

is an Ito process with

dYt =∂g

∂t(t,Xt) dt +

∂g

∂x(t,Xt) dXt +

1

2

∂g2

∂x2(t,Xt) dXt dXt

Proof. A proof of Ito’s lemma can be found on page 46 of [20]

A major component of the theory of stochastic optimal control is the infinitesimal

generator of an Ito diffusion. This can be thought of as and extension of the derivative

from the deterministic calculus, where the limit definition has a similar form with

extensions to include to stochastic nature of an Ito diffusion.

Definition 1.8. Let (Xt)t≥s be an Ito diffusion in R of the form

dXt = α(t,Xt) dt + σ(t,Xt) dBt Xs = x

then the infinitesimal generator A of Xt is defined by

Aϕ(s, x) = limt→s

E s,x [ϕ(t,Xt)]− ϕ(s, x)

t − s

for all x ∈ R, s ∈ [0,∞) and ϕ ∈ C 1,20 ([0,∞),R). The subset of L2([0,∞) × R) for which

the limit Aϕ(s, x) exists for all s, x will be called DA.

It turns out that this ”derivative” in the stochastic sense is related to the classical

derivative in an inherent way, which can be found by applying Ito’s Lemma. The following

theorem shows that the infinitesimal generator of an Ito diffusion turns out to be a

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second order differential operator in the case of diffusions driven by Brownian motion.

It will also be shown later that in the case of diffusions driven by a more general Levy

process the infinitesimal generator will end up being an integro-differential operator.

Theorem 1.3. Let Xt be an Ito diffusion of the form


and consider the differential operator A on C 1,20 ([0,∞),R) given by

A = ∂

∂t+ α

∂

∂x+1

2σ2

∂2

∂x2

and let ϕ ∈ C 1,20 ([0,∞),R) be such that for all t ≥ s, s ∈ [0,∞) and x ∈ R

E s,x[∫ ts

|Aϕ(r ,Xr)dr |]< ∞, and E s,x

[∫ ts

(∂ϕ

∂x(r ,Xr)σ(r ,Xr)

)2dr

]< ∞

then ϕ ∈ DA and Aϕ(s, x) = Aϕ(s, x).

Proof. Since ϕ ∈ C 1,20 ([0,∞),R) we may apply Ito’s formula to compute

dϕ(t,Xt) =∂ϕ

∂t(t,Xt) +

∂ϕ

∂x(t,Xt)dXt +

1

2

∂2ϕ

∂x2(t,Xt)dXt dXt

=∂ϕ

∂t(t,Xt) +

∂ϕ

∂x(t,Xt) [α(t,Xt) dt + σ(t,Xt) dBt ] +

1

2σ2(t,Xt)

∂2ϕ

∂x2(t,Xt)dt

= Aϕ(t,Xt) dt +∂ϕ

∂x(t,Xt)σ(t,Xt) dBt

Integration of both sides of the equation with respect to the proper measures we find

that

ϕ(t,Xt)− ϕ(s,Xs) =

∫ ts

Aϕ(r ,Xr) dr +

∫ ts

∂ϕ

∂x(r ,Xr)σ(r ,Xr) dBr

using the fact that Xs = x and taking the expectation E s,x of both sides of the equation

we are able to compute the numerator of the limit

E s,x [ϕ(t,Xt)]− ϕ(s, x) = E s,x[∫ ts

Aϕ(r ,Xr) dr

]+ E s,x

[∫ ts

∂ϕ


]

18

Given that the function ∂ϕ∂x(r ,Xr)σ(r ,Xr) is B × F-measurable, Fr -adapted, and

E s,x[∫ ts

(∂ϕ∂x(r ,Xr)σ(r ,Xr)

)2dr]< ∞ a standard result from stochastic calculus gives

that

E s,x[∫ ts

∂ϕ


]= 0

hence we have that


Aϕ(r ,Xr) dr

]dividing both sides by t − s and taking limits gives

Aϕ(s, x) =: limt→s

E s,x [Φ(t,Xt)]−Φ(s, x)t − s

= limt→s

E s,x[∫ tsAϕ(r ,Xr) dr

]t − s

= limt→sE s,x

[1

t − s

∫ ts

Aϕ(r ,Xr) dr

]= E s,x

[limt→s

1

t − s

∫ ts

Aϕ(r ,Xr) dr

]= E s,x

[d

dt

(∫ ts

Aϕ(r ,Xr) dr

)]= E s,x [Aϕ(t,Xt)]

=: E [Aϕ(t,Xt)|Xs = x ]

= Aϕ(s, x)

where the fourth equality follows from dominated convergence theorem which we may

apply using the assumption that E s,x[∫ ts|Aϕ(r ,Xr)dr |

]< ∞ so that the term in the

expectation is essentially bounded.

Given the form of the infinitesimal generator of an Ito diffusion we may use the following

theorem to compute expectations of first exit times of the diffusion from a given region.

We will not explicitly compute any expectations of exit times, however we will need

Dynkin’s formula to prove results about exit times of controlled Ito diffusions.

19

Theorem 1.4. (One dimensional Dynkin’s Formula) Let Xt be a solution of an Ito

diffusion of the form


and τ be a stopping time with E s,x [τ ] < ∞. If ϕ ∈ C 1,20 ([0,∞)× R), then

E s,x [ϕ(τ ,Xτ)] = ϕ(s, x) + E x[∫ τ

s

Aϕ(r ,Xr)dr

]Proof. Since ϕ ∈ C 1,20 ([0,∞)× R) we may apply Ito’s formula as in the proof of theorem

to get


Aϕ(r ,Xr) dr

]+ E s,x

[∫ ts

∂ϕ


]Letting t = τ we have that

E s,x [ϕ(τ ,Xτ)] = ϕ(s, x) + E s,x[∫ τ

s

Aϕ(r ,Xr) dr

]+ E s,x

[∫ τ

s

∂ϕ


]To finish the proof we need to now show that E s,x

[∫ τ

sσ(r ,Xr)

∂ϕ∂x(r .Xr)dBr

]= 0. To

simplify notation let g(r ,Xr) = σ(r ,Xr)∂ϕ∂x(r ,Xr), then we have that g is a bounded

Borel Fr -measurable function i.e. |g| ≤ M for some M > 0. Consider the family of

functions∫ τ∧ksg(r ,Xr) dBr

k, I claim this family is uniformly integrable with respect to

the measure Px . To show this family is uniformly integrable it is enough to apply theorem

C .3 from [20], hence we need to show the following integral is finite,

E s,x

[(∫ τ∧k

s

g(r ,Xr)dBr

)2]= E s,x

[∫ τ∧k

s

g2(r ,Xr) dr

]≤ M2 E s,x [τ ∧ k ] ≤ M2 E 2[τ ] < ∞

where the first equality follows from the Ito’s Isometry. The finiteness of this integral from

the use of a quadratic test function gives us that the family is uniformly integrable so that

limk→∞E s,x

[∫ τ∧k

s

g(r ,Xr)dBr

]= E s,x

[limk→∞

∫ τ∧k

s

g(r ,Xr)dBr

]

20

hence combing the results above we have that

E s,x[∫ τ

s

g(r ,Xr)dBr

]= E s,x

[limk→∞

∫ τ∧k

s

g(r ,Xr)dBr

]= limk→∞E s,x

[∫ τ∧k

s

g(r ,Xr)dBr

]= lim

k→∞E s,x

[∫ ks

χr<τg(r ,Xr)dBr

]= 0

where the last equality follows from the fact that∫ ksχr<τg(r ,Xr)dBr is a martingale, the

result now follows from the fact that Aϕ(s, x) = Aϕ(s, x).

21

CHAPTER 2STATEMENT OF MAIN PROBLEM

2.1 Optimal Control Theory

We have now setup a foundation for our problem, this section will state the main

problem and begin to talk about how we may go about finding a solution using optimal

control theory. To begin this section we give some details on controls and how they are

implemented in our problem.

Definition 2.1. Given a Borel measurable set U ⊂ R a control process is a B × F -

measurable stochastic process (ut)t≥s taking values in U i.e. u : [s,∞)×Ω→ U

Definition 2.2. Given a control process (ut)t≥s a controlled process Xt is a solution of

the stochastic differential equation

dXt = α(t,Xt , ut)dt + σ(t,Xt , ut)dBt Xs = x > 0 (2–1)

where α : R × R × U → R,σ : R × R × U → R and Bt is an one-dimensional Brownian

motion and for all x , y ∈ R, t ≥ s, u, v ∈ U we have

|α(t, x , u)− α(t, y , v)|+ |σ(t, x , u)− σ(t, y , v)| ≤ C1|x − y |+ C2|u − v |

|α(t, x , u)|+ |σ(t, x , u)| ≤ D(1 + |x |+ |u|)

The parameter ut ∈ U in Equation (2–1) is used to control the process Xt , given a Borel

set U ⊂ R we vary the parameter ut ∈ U to control the behavior of the process Xt .

The decision on how to vary the control ut is based on the information up to time t, so

that ut is Ft−measurable. Since the control process depends on the underlying ω we

have amended our definition of an Ito diffusion to include appropriate conditions for the

existence and uniqueness of a controlled diffusion. In order to do this we have set it up

so that the global Lipschitz and linear growth conditions hold for all controls ut ∈ U.

To apply the theory of diffusions we need Equation (2–1) to be an Ito diffusion, this is

possible if we restrict the controls we use to a subset of controls called Markov controls

22

Definition 2.3. Let Xt be a solution to Equation (2–1), a Markov control is a control

which does not depend on the initial state of the system (s, x) but only depends on the

current state of the system at any time t i.e. there exits a function u : Rn+1 → U × Rk

such that u(t,w) = u(t,Xt(w))

The Markov control is important in our study because the restriction to this subset of all

controls allows us to turn the equation governing the wealth process into an Ito diffusion.

The fact that the wealth process is a diffusion makes use of important theorems from

the theory of diffusions possible, without this we would not be able to proceed using our

methodology.

2.2 Dynamic Programming Methodology

Consider the following following controlled process on [0, τG ]

dXt = α(t,Xt , ut) dt + σ(t,Xt , ut) dBt X0 = x0 (2–2)

with given performance function

Ju(x0) = Ex0

[∫ τG

0

f (r ,Xr , ur)dr + g(τG ,XτG )χτG<∞

](2–3)

where τG = inft > 0 : (t,Xt) /∈ G is a stopping time, and the set G ⊂ [0, τG ] × R

is called the solvency set. The continuous function f : [0, τG ] × R × R → R is called

the ”utility function” and the continuous function g : R × R → R is called the ”bequest

function”. We would like to find an optimal control u∗ from a set of admissible controls so

that this performance function is maximized. To find our optimal control we must use the

technique of dynamic programming, as the name suggest we need to consider a family

of optimal control problems from which we will select an optimal control. In order to do

this we need to consider different initial times and states along a given trajectory of the

controlled diffusion. If we were to consider a diffusion Xt starting at x0 as in Equation

(2–2) then an admissible control u is (Ft)t≥0 measurable which means that the controller

has all information about the system up to time t so that Xt is almost surely deterministic

23

under a probability measure P(·|Ft). This means for all states of the system we already

know the initial time point and the initial value at this time point. To be able to use the

method of dynamic programming we must vary the initial time and states of the system

and choose the best possible control from a set of admissible controls. To do this we

consider controlled diffusions of the form

dXt = α(t,Xt , ut) dt + σ(t,Xt , ut) dBt Xs = x (2–4)

where t ∈ [s, τG ] and the performance function has the form

Ju(s, x) = E s,x[∫ τG

s


](2–5)

where τG = inft > s : (t,Xt) /∈ G. Before moving on to the statement of the optimal

control problem we need to discuss exactly what mathematical properties an admissible

control should have. We give the general mathematical formulation of an admissible

control over the set [s, τG ] here, we will use a slightly different set later on when we write

the time homogeneous version of the controlled process.

Definition 2.4. We say that the control process u is admissible and write u ∈ A[s, τG ] if

• u is measurable, (Ft)t≥s-adapted with E s,x[∫ τGs

|ur |2 dr]< ∞

• Equation (2–4) has a unique strong solution (Xt)t≥s for all Xs = x and

E s,x[sups≤t≤τG

|Xt |2]< ∞

• Ju(s, x) is well defined i.e.

E s,x[∫ τG

s


]< ∞

The stochastic control problem may now be stated, given continuous utility function f

and bequest function g we would like to find optimal control u∗ and value function Φ(s, x)

24

such that

Φ(s, x) = supu∈A[s,τG ]

Ju(s, x) = Ju∗(s, x) (2–6)

Φ(τG , x) = g(τG , x) (2–7)

subject to the constraint of Equation (2–4). To begin the analysis of the optimal control

problem in Equation (2–6), we introduce some notation that will make our problem a

time homogeneous control problem. We first begin by rewriting Equation (2–4) using the

substitution

Yt =

s + tXs+t

t ≥ 0

which gives us a time homogeneous controlled process Yt satisfying

dYt = α(Yt , us+t)dt + σ(Yt , us+t)dBs+t Y0 = (s, x) =: y (2–8)

We may also recast the performance function in Equation (2–5) using our new notation,

letting r = s + t we have∫ τG

s

f (r ,Xr , ur)dr =

∫ τG−s

0

f (s + t,Xs+t , us+t)dt =

∫ τG

0

f (Yt , us+t)dt

where τG = τG − s, hence the performance function becomes

Jus+t(y) = E y[∫ τG

0

f (Yt , us+t)dt + g(YτG )χτG<∞

](2–9)

Since dBs+t has the same distribution as dBt we may make this replacement in the

controlled process equation. We note that the control from Equation (2–9) is a time

shifted version of the control from Equation (2–11), after the substitution for Yt we see

that we are now working with the control process us+t in each equation. The solution

to the optimal control problem is found by working in two different worlds, this idea is

at the center of our entire analysis hence we shall spend some time discussing how to

go back and forth between each world. We summarize each of the equations and the

25

vital information in each world in Table 2− 1, this table is very useful when comparing

equations between the different worlds. The main idea of working between the two

Table 2-1. Table with comparison of two worlds used to solve optimal control problemTwo worlds

Original Xt space Transformed Yt = (s + t,Xs+t) spacedXt = α(t,Xt , ut) dt + σ(t,Xt , ut) dBt dYt = α(Yt , us+t) dt + σ(Yt , us+t) dBtXs = x Y0 = (s, x) =: y

J(s, x) = E s,x

[∫ τG

s

f (t,Xt , ut) dt + g(τG ,XτG )

]J(y) = E y

[∫ τG

0

f (Yt , us+t) dt + g(YτG )

]ut = u(t,Xt) us+t = u(s + t,Xs+t) = u(Yt)ut ∈ A[s, τG ] u∗s ∈ A[0, τG ]

worlds is that we will be solving the optimal control problem in the Yt space where the

control is us+t and then we will transform back to the Xt space where the control is ut .

The connection between the two worlds that we will need is the fact that in the Yt world

we have that u∗s = u∗(s, x) = u∗(s,Xs) since Xs = x in the Xt world. Now renaming

the variable we have that the optimal control in the Xt world (denoted simply ut) is given

by ut = u∗(t,Xt) where u∗ is the optimal control we find in the Yt world. Hence, the

optimal control problem is solved by transforming into the Yt world where an optimal

control u∗(s,w) will be found, then using that optimal control we are able to transform

back to the Xt world to solve for the optimal control ut(t,Xt) which is a solution of the

control problem in the Xt world. However for now we make no distinction between the

different versions of α,σ and u and state the control problem in the new notation. Given

utility function f and bequest function g we would like to find optimal control u∗ and value

function Φ(y) such that

Φ(y) = supu∈A[0,τG ]

Ju(y) = Ju∗(y) (2–10)

where

J(y) = E y[∫ τG

0

f u(Yt , us+t)dt + g(YτG )χτG<∞

](2–11)

26

subject to the constraint

dYt = α(Yt , us+t)dt + σ(Yt , us+t)dBt Y0 = (s, x) =: y (2–12)

In our new notation a Markov control (we make no distinction between u and u) is one

such that us+t = u(s + t,Xs+t) = u(Yt) hence we may rewrite Equation (2–12) as

dYt = α(Yt , u(Yt))dt + σ(Yt , u(Yt))dBt Y0 = (s, x) =: y (2–13)

We now have the right hand side of Equation (2–13) only depends on the state of

the system Yt at time t and does not depend on time explicitly, i.e. the solution Yt

of Equation (2–13) is a time homogeneous Ito diffusion. Given that the state of our

system has this form we may now apply the theory of diffusions to solve our optimal

control problem. Before stating the main theorems necessary to solve the optimal

control problem we mention that the notation between the two worlds will be used

interchangeably for the remainder of the paper, any confusion with the notation can be

resolved by consulting Table 2− 1. The main theorems of the paper will be stated in the

Yt world since much of the work will be done in this setting and the results will then be

transformed back to the Xt world to provide a solution to the original problem.

2.3 Hamilton Jacobi Bellman

The Hamilton Jacobi Bellman theorem is at the heart of much of the analysis that

will be performed throughout the paper, it use will be of an indirect nature but it is of

great importance hence we provide the complete statement and proof.

Theorem 2.1. Let Ju(y) be as in Equation (2–11), where u = u(Y ) is a Markov control

and

Φ(y) = supuJu(y)

27

Supposing the function Φ ∈ C 2G ∩ C(G) is bounded for all finite stopping times τG a.s.

Py and all y ∈ G . If an optimal Markov control u∗ exists and ∂G is regular for Y u∗t then

supu∈A[0,τG ]

[f u(y) + AuΦ(y)] = 0 for all y ∈ G (2–14)

and

Φ(y) = g(y) for all y ∈ ∂G (2–15)

Proof. Since ∂G is regular for Y u∗t we have τG = 0 for any y ∈ ∂G , so that

Φ(y) = g(YτG )χτG<∞ = g(y)

hence Equation (2–15) holds for all y ∈ ∂G . To prove Equation (2–14) fix y = (s, x) ∈ G

and take u to be any Markov control. If τ ≤ τG is a stopping time then we may compute

E y [Ju(Yτ)] = E y[EYτ

[∫ τG

0

f u(Yr) dr + g(Yτ)

]]= E y

[E y[∫ τG

τ

f u(Yr) dr + g(Yτ)|Fτ

]]= E y

[E y[∫ τG

0

f u(Yr) dr + g(YτG )−∫ τ

0

f u(Yr) dr |Fτ

]]= E y

[∫ τG

0

f u(Yr) dr + g(YτG )−∫ τ

0

f u(Yr) dr

]= E y

[∫ τG

0

f u(Yr) dr + g(YτG )

]− E y

[∫ τ

0

f u(Yr) dr

]= Ju(y)− E y

[∫ τ

0

f u(Yr) dr

]This shows that

Ju(y) = E y[∫ τ

0

f u(Yr) dr

]+ E y [Ju(Yτ)] (2–16)

which is an equality used to prove Bellman’s principle of optimality which we do not

directly prove here, but we will use the above equality to finish the proof. First we must

define the proper continuation region, let U ⊂ G of the form U = (r , x) ∈ G : r < t1

28

where s < t1. Then if τ = τU is the first exit time from U and u∗(y) = u∗(r , x) is the

optimal control, defining a control

u(r , x) =

v if (r , x) ∈ U

u∗(r , x) x ∈ G\U

where v ∈ A[0, τG ] is an arbitrary control. With this continuation region we have that

Φ(Yτ) = Ju∗(Yτ) = J

u(Yτ) (2–17)

Using the fact that ϕ(y) is the supremum and Equation (2–16) we have

Φ(y) ≥ Ju(y) = E y[∫ τ

0

f u(Yr) dr

]+ E y [Ju(Yτ)]

= E y[∫ τ

0

f u(Yr) dr

]+ E y [ϕ(Yτ)] (2–18)

Since we assumed Φ(y) ∈ C 2(G) and τ is a stopping time we may apply Dynkin’s

formula to get

E y [Φ(Yτ)] = Φ(y) + Ey

[∫ τ

0

AuΦ(Yr)dr

](2–19)

If we now plug Equation (2–19) into Equation (2–18) we get

Φ(y) ≥ E y[∫ τ

0

f u(Yr) dr

]+Φ(y) + E y

[∫ τ

0

AuΦ(Yr) dr

]Combining expectation and subtracting Φ(y) from both sides we have

E y[∫ τ

0

f u(Yr) dr + AuΦ(Yr) dr

]≤ 0

Now we let t1 → s and using the fact that f (·) and AuΦ(·) are continuous at y we may

perform the integration to get that

(f u(y) + AuΦ(y))E y [τ ] ≤ 0

29

dividing out E y [τ ] we have that f u(y) + AuΦ(y) ≤ 0 for all stopping times u ∈ A[0, τG ].

The supremum is obtained at the optimal control u∗ where

Φ(y) = Ju∗(y) = E y

[∫ τG

0

f u∗(Yr )(Yr) dr + g(YτG )

](2–20)

is a solution of the combined Dirichlet-Poisson problem so that

Au∗(y)Φ(y) = −f u∗(y)(y) for all y ∈ G

This theorem gives a necessary condition that states if an optimal control u∗ exists then

the function F (u) = f u(y) + Auϕ(y) attains it maximum value 0 at u = u∗. However

it does not address the question of sufficiency of the optimal control u∗ i.e. if for every

point y ∈ G we find a u∗(y) such that F (u∗(y)) = 0, will u∗(y) be an optimal control?

This question can be answered by the following converse

Theorem 2.2. Let ϕ ∈ C 2(G) ∩ C(G) satisfy the following conditions

• f u(y) + Auϕ(y) ≤ 0 y ∈ G and for all u ∈ A[0, τG ]

• limt→τG

ϕ(Yt) = g(YτG )χτG<∞ a.s.Py

• ϕ(Yτ)τ≤τG is uniformly Py integrable for all Markov controls and all y ∈ G .

then ϕ(y) ≥ Ju(y) for all Markov controls u ∈ A and y ∈ G . Moreover if for all y ∈ G we

find a Markov control u = u0(y) such that

f u0(y)(y) + Au0(y)ϕ(y) = 0 (2–21)

then u∗t = u0(Yt) is optimal and ϕ(y) = Φ(y) = Ju∗(y).

Proof. Let R < ∞ and define the stopping time

TR = minR, τG , inft > 0; |Yt | ≥ R

30

so that we have limR→∞

TR = τG . We may apply Dynkin’s formula and use the hypothesis

that Auϕ(y) ≤ −f u(y) for all y ∈ G to get that

E y [ϕ(YTR )] = ϕ(y) + E y[∫ TR0

Auϕ(Yr) dr

]≤ ϕ(y)− E y

[∫ TR0

f uϕ(Yr) dr

]Rearranging of this equation and combining terms with expectations gives us that

ϕ(y) ≥ E y[∫ TR0

f u(Yr) dr + ϕ(YTR )

](2–22)

Applying Fatou’s Lemma to this equation and using the remaining hypothesis we have

ϕ(y) ≥ lim infR→∞

E y[(∫ TR

0


)]≥ E y

[lim infR→∞

(∫ TR0


)]= E y

[∫ T0

f u(Yr) dr + limR→∞

ϕ(YTR )

]= E y

[∫ T0

f u(Yr) dr + limt→τG

ϕ(Yt)

]= E y

[∫ T0

f u(Yr) dr + g(YτG )χτG<∞

]= Ju(y)

To complete the proof, if we are able to find a control u0(y) such that f u0(y)(y) +

Au0(y)ϕ(y) = 0 then we may apply the same argument as above to ut = u0(Yt) so

that the inequality becomes equality i.e. ϕ(y) = Ju∗(y) and u∗ is an optimal control.

2.4 Verification Theorem

The Hamilton Jacobi Bellman equation has many technical conditions that are

generally difficult to check explicitly, in particular we do not know the solution Φ a priori.

Each version of the Hamilton Jacobi Bellman equation requires that we know that the

solution satisfy Φ ∈ C 2(G) ∩ C(G). This makes solving the optimal control problem

using the Hamilton Jacobi Bellman theorems directly rather difficult, in order to solve

31

this problem we introduce the idea of verification theorems. Using the Hamilton Jacobi

Bellman theorems would require us to know the solution Φ and check all the required

technical conditions, once this is done then we are able to determine if a control u is an

optimal control and show that Φ is actually a solution to the Hamilton Jacobi Bellman

equation. The verification theorem reverse the order of this process. We first begin with

a solution ϕ to the Hamilton Jacobi Bellman equation from which we are able to also

get a candidate for the optimal control u∗. Once we have shown that ϕ satisfies the

conditions of the verification theorem we can then show that ϕu∗ = Φ so that the optimal

control problem is solved.

Theorem 2.3. Let u ∈ A[s, τG ] and (s, x) ∈ G and suppose the following conditions are

satisfied for all s ∈ [0, τG ] and x ∈ R

• ϕ ∈ C 1,2([0, τG)× R) is continuous on [0, τG ]× R and satisfies the quadratic growthcondition |ϕ(s, x)| ≤ Cϕ(1 + |x |2)

• ϕ satisfies the Hamilton Jacobi Bellman equation

supu∈A[s,τG ]

[f u(s, x) + Auϕ(s, x)] = 0 s ∈ [0, τG)

ϕ(τG , x) = g(τG , x)

• f u is continuous with |f u(s, x)| ≤ Cf (1 + |x |2 + ||u||2) for some constant Cf > 0.

• |σu(s, x)|2 ≤ Cσ(1 + |x |2 + ||u||2) for some constant Cσ > 0.

then ϕ(s, x) ≥ Φ(s, x) for all (s, x) ∈ G . Moreover if uo(s, x) is the max of u 7→

f u(s, x) + Auϕ(s, x) and u∗ = u0(s,Xs) is admissible then ϕ(s, x) = Φ(s, x) for all

(s, x) ∈ G and u∗ is and optimal strategy i.e. Φ(s, x) = Ju∗(s, x).

Proof. Fix s ∈ [0, τG ] and x ∈ R, to get that the process X is bounded we define the

stopping time

τn = τG ∧ inft > s : |Xt − Xs | ≥ n

Let u ∈ A[s, τG ] be an admissible control and Xs = x then by Ito’s lemma we have that

ϕ(τn,Xτn) = ϕ(s, x) +

∫ τn

s

Aurϕ(r ,Xr) dr +

∫ τn

s

ϕx(r ,Xr)σur (r ,Xr) dBr (2–23)

32

Since ϕx(s,Xs) is continuous on the set [s, τn] we have that there exists a constant Cϕx

such that |ϕx(s, x)|2 ≤ Cϕx . Using the fact that ur is admissible and that Xr is bounded on

[s, τn] we find that

E s,x[∫ τn

s

|ϕx(r ,Xr)σur (r ,Xr)|2 dr]≤ E s,x

[∫ τn

s

|ϕx(r ,Xr)|2|σur (r ,Xr)|2 dr]

≤ E s,x[∫ τG

s

CϕxCσ(1 + |Xr |2 + ||ur ||2) dr]< ∞

hence∫ τns

ϕx(r ,Xr)σur (r ,Xr) dBr is a martingale which gives that

E s,x[∫ τn

s

ϕx(r ,Xr)σur (r ,Xr) dBr

]= 0

Taking expectations on both sides of Equation (2–23) and using the last calculation we

get that

E s,x [(τn,Xτn)] = Es,x

[ϕ(s, x) +

∫ τn

s

Aurϕ(r ,Xr) dr

]adding E s,x

[∫ τnsf ur (r ,Xr) dr

]to both sides of this equation and combining terms into a

single expectation we find that

E s,x[∫ τn

s

f ur (r ,Xr) dr + ϕ(τn,Xτn)

]= E s,x

[ϕ(s, x) +

∫ τn

s

(f ur (r ,Xr) + Aurϕ(r ,Xr)) dr

]≤ E s,x [ϕ(s, x)] = ϕ(s, x) (2–24)

where we have used that fact that for any control ur ∈ A we have that f ur (r ,Xr) +

Aurϕ(r ,Xr) ≤ 0. If we can now show that the left hand side goes to Ju(s, x) as n → ∞

the proof will be complete. In order to do this we need to let n → ∞ so that τn → τG ,

our main concern here will be bringing the limit inside of the expectation operator. To

accomplish this we would like to used dominated convergence theorem, hence we need

to check that the hypothesis are met i.e.∣∣∣∣∫ τn

s


∣∣∣∣ ≤ ∫ τn

s

|f ur (r ,Xr)| dr + |ϕ(τn,Xτn)|

≤ Cf∫ τG

s

(1 + |Xr |2 + ||ur ||2) dr + Cϕ(1 + |XτG |2) ∈ L1

33

an application of dominated convergence gives and the fact that ϕ(τG ,XτG ) = g(τG ,XτG )

E s,x[∫ τn

s


]−→ E s,x

[∫ τG

s

f ur (r ,Xr) dr + g(τG ,XτG )

]= Ju(s, x)

Letting n → ∞ on both sides of Equation (2–24) and using this result gives us that

Ju(s, x) ≤ ϕ(s, x) for all u ∈ A

since the right hand side does not depend on u we take the supremum of both sides

over the set of all admissible controls to get

Φ(s, x) = supu∈AJu(s, x) ≤ ϕ(s, x)

If we were now able to find a maximizer u∗ the only part of the proof above that would

be different is the fact that instead of an inequality in the argument we would have

equality in Equation (2–24) so that we may continue the argument to get that Φ(s, x) =

Ju∗(s, x) = ϕ(s, x) completing both parts of the proof.

Hence the verification theorem gives us a way of finding a solution to the optimal control

problem without having to check the complicated hypotheses of the Hamilton Jacobi

Bellman equations. The verification methodology of solving the optimal control problem

using the verification theorem is as follows

• Write down the Hamilton Jacobi Bellman equation and the appropriate boundarycondition.

• Take the first order derivatives with respect to the control variables in the HamiltonJacobi Bellman equation, and solve for the optimal control candidate u0.

• Plug the optimal controls back into the Hamilton Jacobi Bellman equation to get anon linear partial differential equation which has to be solved for a candidate of thevalue function ϕ subject to the boundary condition

• Show that ϕ satisfies the conditions of the verification theorem and that u∗s =u0(s,Xs) is an admissible control for all s ∈ [0, τG ], then ϕ = Φ and u∗ is an optimalcontrol strategy.

34

CHAPTER 3MERTON PROBLEM

3.1 Classic Merton Solution

This section formulates and solves a version of the classic optimal control problem

that Merton solved in his 1971 paper [18]. In his paper Merton assumed a general family

of utility functions called the Hyperbolic absolute risk aversion utilities. I will present a

subset of this family which avoids the complete generality as Merton originally presented

but which still captures the essence of his results. The original problem considers a

portfolio of n assets, however with the assumption of log normally distributed assets

we may use a so called mutual fund theorem which allows us to instead consider a

portfolio of 2 assets. The assets under consideration are a risk free asset (such as a

U.S. treasury bond) and a risky asset (such as a share of stock) which may be written

as a linear combination of the n assets. To begin the analysis let (Ω,F ,F[0,∞),P) be

a filtered probability space. The risk free Rt asset evolves according to the differential

equation

dRt = rRt dt; R0 = 1 (3–1)

where r ≥ 0 is a constant which represents the risk free rate of interest, while the risky

asset St evolves according to a Geometric Brownian motion

dSt = αSt dt + σSt dBt ; S0 = s0 (3–2)

where α ≥ 0 and σ ≥ 0 represent the rate of the return and volatility of the asset St

respectively.

Definition 3.1. (Trading strategy)A trading strategy is a two dimensional stochastic

process πt = (π0t , π1t )t∈[0,∞), such that πt is B × F-measurable and Ft-adapted.

Financially we interpret π0t as the number of shares in the risk free asset Rt , while π1t is

the number of shares in the risky asset St at time t. Hence the total wealth at time t of

35

the portfolio of assets may be written as

Wt = π0tRt + π1t St (3–3)

Let ct be an Ft adapted process that the investor is able to choose at time t, which

represents the rate at which money can be moved from the risk free asset to the risky

asset within the portfolio.

Definition 3.2. Self financing A trading strategy (π0t , π1t ) is called self financing if the

corresponding wealth process (Wt)t≥0 is continuous and adapted such that

Wt = w0 +

∫ t0

π0udRu +

∫ t0

π1udSu −∫ t0

cu du (3–4)

The assumption of a self financing portfolio of assets basically states that no sources of

external capital can be added to the portfolio, any capital gains must be reinvested into

the portfolio. We also have underlying assumptions that there are no transaction costs

to redistribute capital between the two assets. Let θt be the fraction of total wealth of the

portfolio invested in the risky asset, then we may write

θt =π1t StWt

while the fraction of wealth in the risk free asset is π0t RtWt

. Assuming thatWt > 0 for all t

we may divide Equation (3–5) byWt to see that 1−θt =π0t RtWt

. With these assumption and

the notation we above we may write the change is the wealth process using Equation

(3–4) as

dWt = π0t dRt + π1t dSt − ct dt

= rπ0tRt dt + π1t [αSt dt + σSt dBt ]− ct dt

= (1− θt)rWt dt + αθtWt dt + σθtWtdBt − ct dt

= (α− r)θtWtdt + [rWt − ct ]dt + σθtWtdBt (3–5)

36

Equation (3–5) along with initial conditionW0 = π00R0 + π10S0 =: w will serve as the

controlled diffusion for the optimization problem. The control process in this problem

is given by the vector ut = (θt , ct), where θt is the fraction of wealth in the risky asset

and ct is the consumption process. The freedom of choice for θt can be thought of

as an investor choosing the fraction of total wealth of the portfolio he/she would like

invested in the risky asset. At each point in time the investor much choose this fraction

along with the consumption process ct as to maximize some performance function.

The reallocation of portfolio weights θt at time t is the continuous time version of the

problem solved by Henry Markowitz [17], which was done in a discrete time setting

and is famously referred to as Modern Portfolio Theory. We will assume a trader has

complete information in the market up to time t so that the control process ut = (θt , ct)

are adapted with respect to the standard filtration. We may now define the value function

for the optimization problem in the classical case, we look at several cases. In this

chapter we will consider cost functionals on an infinite horizon with utility functions of the

form f (t,Wt , ut) = e−δtU(ct) where there is no explicit dependence onWt . However,

since ct is a stochastic process there will be a built in dependence onWt which will

show up when we find the control process explicitly. Given this form of the utility function

the cost functional becomes

Jθ,c(s,w) = E s,w[∫ ∞

s

e−δtU(ct) dt

]= e−δsE y

[∫ ∞

0

e−δtU(cs+t) dt

](3–6)

where the (utility) function U is increasing, differentiable and concave in ct . The

process ct is the consumption rate that the investor must choose when rebalancing the

portfolio so as to maximize the cost functional. So the optimal control problem we are

considering in this section is one in which the investor tries to maximize the expected

utility of consumption over the given trading period by choosing the appropriate controls.

37

In mathematical terms we would like to find Φ(s,w) such that

Φ(s,w) = supθ,cJθ,c(s,w) = sup

θ,cE s,w

[∫ ∞

s

e−δtU(ct) dt

](3–7)

To see how the value function ϕ(s,w) behaves in each of the two worlds we perform the

following calculation where the control process is ut = (θt , ct)

Φ(s,w) = supθ,cE s,w

[∫ ∞

s

f (t,Wt , ut) dt

]= sup

θ,cE s,w

[∫ ∞

s

e−δtU(ct) dt

]= sup

θ,cE s,w

[∫ ∞

0

e−δ(s+r)U(cs+r) dr

]= sup

θ,cE s,w

[∫ ∞

0

f (s + r ,Ws+r , us+r dr

]= sup

θ,cE y[∫ ∞

0

f (Yr , us+r) dr

]= sup

θ,cE y[∫ ∞

0

f (Yr , u(Yr)) dr

]= Φ(y)

The equality before the last comes from the fact that we will be using Markov controls

throughout our analysis, hence we have that us+r = u(s + r ,Ws+r) = u(Yr). This

argument provides the glue to bind together the optimal control problem between the Xt

and Yt spaces. In particular we are able to find the optimal controls using the Hamilton

Jacobi Bellman equation in the Yt space, then transform back into the Xt space to find

the optimal control process. It also shows that the value functions in each of the two

spaces will be the same as Φ(s,w) = Φ(y), hence we will use these interchangeably

from this point on. To solve the optimal control problem we will perform most of our

analysis in the Yt space, once the problem is solved in this space we will only transform

back to the Xt space the find the optimal control process ut . In particular the Ito diffusion

and Hamilton Jacobi Bellman equation will live in the Yt space, once the optimal controls

are found we will use the verification theorem to write the controls in the Xt space.

3.2 Infinite Horizon Power Utility

The first optimal control problem we consider is equivalent to the control problem

considered by Merton in [18], in particular we consider the utility function f (t,Wt , ut) =

e−δt cγt

γ. Merton solves this problem in a more general setting, but we will consider only a

special case of the solution so that we may make direct comparisons with the remainder

38

of the paper. The cost functionals in both the Xt and Yt space are given by

Jθ,c(s,w) = E s,w[∫ ∞

s

e−δt cγt

γdt

]= e−δsE y

[∫ ∞

0

e−δt cγs+t

γdt

](3–8)

Given the cost functional in the Yt space we would like to find optimal controls c∗, θ∗ and

value function Φ(s,w) such that

Φ(s,w) = supθ,cJθ,c(s,w) = Jθ

∗,c∗

In order to find the optimal controls and the value function, we need to apply the

verification theorem. To do this we will assume that ϕ is bounded and optimal controls

c∗ and θ∗ exists, then the value function must satisfy the the Hamilton Jacobi Bellman

equation for the Ito diffusion in Equation (3–5) for the wealth process i.e. ϕ satisfies

supθ,c

[e−δs c

γ

γ+ ϕs + θ(α− r)wϕw + (rw − c)ϕw +

1

2σ2w 2θ2ϕww

]= 0 (3–9)

The first order optimal conditions are found by taking first derivatives with respect to c

and θ and solving for the critical points respectively so that the possible optimal controls

are

c∗ =(eδsϕw

) 1γ−1 θ∗ =

−(α− r)ϕwσ2wϕww

(3–10)

With these optimal controls the Hamilton Jacobi Bellman equation becomes

e−δs

γ

(eδsϕw

) γγ−1 + ϕs −

(α− r)2ϕ2wσ2ϕww

+ rwϕw −(eδsϕw

) 1γ−1 ϕw +

(α− r)2ϕ2w2σ2ϕww

= 0

=⇒ (eδs)1

γ−1

γϕ

γγ−1w + ϕs −


+ rwϕw −(eδs) 1

γ−1 ϕγ

γ−1w = 0

=⇒ 1− γ

γ(eδs)

1γ−1ϕ

γγ−1w + ϕs −


+ rwϕw = 0

Hence we have that our optimal control problem is reduced to solving this partial

differential equation. By the argument above we know that the value function has

the form ϕ(s,w) = e−δsϕ(0,w), it was shown by Merton and others since then that

39

ϕ(0,w) = K p0wγ where K p0 is a constant so that ϕ(s,w) = K p0 e−δsw γ. It turns out that

we may find the value of the constant K0 explicitly. To this end we find the necessary

derivatives and plug into the equation we find that our equation becomes

1− γ

γ(eδs)

1γ−1 (K p0 γ)

γγ−1w γ − K p0 δe−δsw γ − (α− r)2γK p0 e−δs

2σ2(γ − 1)w γ + rγK p0 e

−δsw γ = 0

=⇒ K p0 e−δs

[1− γ

γ(K p0 )

1γ−1 (γ)

γγ−1 − δ − (α− r)2γ

2σ2(γ − 1)+ rγ

]w γ = 0

=⇒ K p0 = 0 (K p0 γ)1

γ−1 =1

1− γ

(δ − γr +

(α− r)2γ2σ2(γ − 1)

)Taking the non trivial solution we find that

K p0 =1

γ

[1

1− γ

(δ − γr − (α− r)2γ

2σ2(1− γ)

)]γ−1Hence the solution to the optimal control problem is given by

ϕ(s,w) =1

γ

[1

1− γ

(δ − γr − (α− r)2γ

2σ2(1− γ)

)]γ−1e−δsw γ (3–11)

This solution is only valid when

δ > γ

[r +

(α− r)2

2σ2(1− γ)

](3–12)

so that the term in brackets is positive and we have a real solution. Choosing parameters

so that this condition is satisfied we present a graph of the value function in Figure 3− 1,

we will use this graph to compare to the other cases and to verify the consistency of our

results. In this case we are able to find the optimal controls explicitly since the value

function is is smooth enough. The explicit solutions are found by plugging ϕ(s,w) back

into Equation (3–10) to get

c∗ =(eδsϕw

) 1γ−1 = K p0w (3–13)

40

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

Infinitemax= 4.676666min= 0.5440538

Figure 3-1. Value function for power utility with no jumps with parameters,α = .16, r = .06, δ = 1, γ = .5, σ = .3

for the optimal consumption rate and

θ∗ =−(α− r)ϕwσ2wϕww

=α− r

σ2(1− γ)(3–14)

Hence we have that the consumption rate ct = K p0Wt is a linear function of the wealth

process. Since K p0 ≥ 0, intuitively this means that an increase in wealth process should

lead to an increase in consumption by the investor. The fraction of wealth in the risky

asset θt = α−rσ2(1−γ)

turns out to be a constant in time, which means that the investor

should continually change his/her portfolio weight throughout the investment period

so that the weight is equal to this constant to optimize the given utility function. This is

difficult to do in a real world setting as it is difficult to continuously execute trades to keep

the portfolio weight constant, but is useful in the sense that the investor knows exactly

how to choose his/her consumption as time evolves through the investment period.

In actuality the investor would execute trades at a finite set of times, trying to keep

the weight of the risky asset as close to constant θt as possible to get the best result.

The existence of the optimal controls and the solution of the optimal control problem

all depends on the stochastic differential Equation (3–5) have a solution. With explicit

41

solutions for the consumption and fraction of the risky asset given by

ct = Kp0Wt θt =

α− rσ2(1− γ)

(3–15)

we may plug these into Equation (3–5) to see why a solution is possible and we may

also find the explicit solution if necessary,

dWt = (α− r)θtWt dt + [rWt − ct ]dt + σθtWt dBt

=(α− r)2

σ2(1− γ)Wtdt + rWt dt − KP0 Wt dt +

α− rσ(1− γ)

Wt dBt

=

[(α− r)2

σ2(1− γ)+ r − KP0

]Wt dt +

α− rσ(1− γ)

Wt dBt

Hence the wealth process is a Geometric Brownian motion of the form

dWtWt=

[(α− r)2

σ2(1− γ)+ r − KP0

]dt +

α− rσ(1− γ)

dBt Ws = w (3–16)

whose solution exists and is unique givenWs = w . Equation (3–16) is a Geometric

Brownian motion whose solution is given by

Wt = w e

([(α−r)2

σ2(1−γ)+r−KP0

]− 12(α−r)2

σ2(1−γ)2

)(t−s)+ α−r

σ(1−γ)

√t−sN(0,1)

(3–17)

where N(0, 1) is a standard normal distribution. Given this form and values for the

parameters we have simulated this wealth process and the result is given in Figure

3− 2. The interpretation of the constant value of θ is that for the investor optimize the

cost function they must maintain this constant fraction of wealth in the risky asset over

the investment period. For a more precise mathematical interpretation we recall that the

fraction of wealth in the risky asset is given by

θt =π1t St

π0tRt + π1t St=

α− rσ2(1− γ)

=: θ0

42

Total Wealth for Power Utility

Time

W_t

0.0 0.2 0.4 0.6 0.8 1.0

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Figure 3-2. Wealth process for power utility with no jumps with parameters,w = 1,α = .16, r = .06, δ = 1, γ = .5, σ = .3

for the investor to keep this fraction of total wealth equal to the constant θ∗ he/she must

satisfy the following condition

π1t St =θ01− θ0

π0tRt

This means the investor must keep his/her wealth invested in the risky equal to a

constant linear multiple of the wealth invested in the risk free asset. Hence the investor

must constantly rebalance the trading strategy (π0t ,π1t ) if they hope to maximize the

expected utility of consumption over the investment period. The continual rebalancing of

the positions of the asset in the portfolio requires a consideration of transaction costs as

this will surely be a factor in how often the investor is willing to perform the rebalancing.

This is a matter that has been studied in papers such as [10] and [21] and one which

we do not address here, one final step remains to completely classify the solution and

that is the issue of checking the conditions of the verification theorem. To show that

the solution ϕ(s,w) is optimal we need to show that the conditions of the verification

theorem are satisfied so that the methodology is legitimate for the solution. We first need

to show that the control process ut = (ct , θt) is an element of A[s,∞) to make sure that

43

we have an admissible control. The precise form of the control process is

ut =

(KP0 Wt ,

α− rσ2(1− γ)

)which is measurable and Ft-adapted by inspection. We next check that the control

process satisfies the following integrability condition

E s,w[∫ ∞

s

||ut ||2 dt]= E s,w

[∫ ∞

s

(K p0Wt)

2 +

(α− r

σ2(1− γ)

)2dt

]

= (K p0 )2E s,w

[∫ ∞

s

W 2t dt

]+

(α− r

σ2(1− γ)

)2Ps,w [s,∞) < ∞

The first term is finite sinceWt is a unique strong solution, and the second terms is finite

since we are working on a probability space. Hence the control process in this case is

appropriately behaved as far integrability in the L2 sense goes. The final condition for

admissibility is to check the finiteness of the cost functional which only depends on the

fact that the optimal consumption process satisfies ct = K p0Wt , hence this gives us that

Jθ,c(s,w) = E s,w[∫ ∞

s

e−δt (Kp0Wt)

γ

γdt

]≤ (K p0 )

γ

γE s,w

[∫ ∞

s

W γt dt

]≤ (K p0 )

γ

γE s,w

[∫ ∞

s

(1 +W 2t ) dt

]≤ (K p0 )

γ

γ

(Ps,w [s,∞) + E s,w

[∫ ∞

s

W 2t dt

])< ∞

Hence the control process ut ∈ A[s,∞), to finish the verification of the technical

conditions we must verify that the hypotheses of the verification theorem are satisfied.

First we note that ϕ(s,w) ∈ C 1,20 , and satisfies the Hamilton Jacobi Bellman equation by

construction of the constant K p0 . The growth condition also holds since

|ϕ(s,w)| = |eδsK p0w γ| ≤ K p0 |w |γ ≤ K p0 (1 + |w |2)

44

Using a similar calculation for f c,θ(s,w) we may show that

|f c,θ(s,w)| =∣∣∣∣e−δs c

γ

γ

∣∣∣∣ ≤ γ−1|c |γ ≤ γ−1(1 + |c |2)

and the condition for σc,θ(s,w) satisfies

|σc,θ(s,w)|2 = |σθw |2 = σ2θ2|w |2 ≤ σ2θ2(1 + |w |2)

The calculations allow us to use the verification Theorem 2.3 for this case of the optimal

control problem. Hence an application of this theorem shows that the candidate ϕ(s,w)

is actually the optimal solution to the control problem i.e. ϕ(s,w) = Φ(s,w). For the

remainder of the paper the check of the technical conditions will be less rigorous than

this section, as many of the arguments are similar to the above routine. The power utility

function considered in this section will be the main focus of the paper, however there

is a special case of the family of hyperbolic absolute risk aversion utility functions that

provide closed form solutions in the infinite horizon case. This special case is that of

logarithmic utility of which we will consider in the following section.

3.3 Infinite Horizon Log Utility

The second case we consider in the infinite horizon case is that of a log utility

function i.e. U(c) = log c . This is another example of a utility function from the

hyperbolic absolute risk aversion family which exhibits decreasing absolute risk aversion,

the cost functional has the form

Jθ,c(s,w) = E s,w[∫ ∞

s

e−δt log(ct) dt

]= e−δsE y

[∫ ∞

0

e−δt log(cs+t) dt

](3–18)

In this case the Hamilton Jacobi Bellman equation only changes in the first term

supθ,c

[e−δs log c + ϕs + θ(α− r)wϕw + (rw − c)ϕw +

1

2σ2w 2θ2ϕww

]= 0

Since we are working over an infinite horizon the value function has the form ϕ(s,w) =

e−δs ϕ(w), so we may rewrite the Hamilton Jacobi Bellman equation without explicit

45

dependence on the time variable

supθ,c

[log c − δϕ+ θ(α− r)w ϕw + (rw − c)ϕw +

1

2σ2w 2θ2ϕww

]= 0 (3–19)

hence the first order condition for θ∗ changes only slightly while the condition for c∗ is

different and is given by

c∗ =1

ϕwθ∗ =

−(α− r)ϕwσ2w ϕww

(3–20)

Trying a function of the form ϕ(w) = (K l0 logw+Kl1) and plugging these into the Hamilton

Jacobi Bellman equation we find that

log

(1

ϕw

)− δϕ− (α− r)2ϕ2w

σ2ϕww+ rw ϕw − 1 + (α− r)2ϕ2w

2σ2ϕww= 0

=⇒ log(1

ϕw

)− δϕ+


+ rw ϕw − 1 = 0

taking necessary derivatives and plugging into Equation (3–19) gives

(1− δK l0) logw +

(− log(K l0)− δK l1 + K

l0

(α− r)2

2σ2+ rK l0 − 1

)= 0

Noting that the term in parentheses is constant in w gives that the sum is zero only if

the coefficient on the logw is zero as well (Since the constant term and the log term are

independent of each other). The constant of the log term being zero gives

1− δK l0 = 0 =⇒ K l0 =1

δ

while setting the constant term equal to zero and solving for K l1 gives

K l1 =1

δ

[− log(K l0) + K l0

(α− r)2

2σ2+ rK l0 − 1

]Using the fact that K l0 =

1δ

we may find the other constant is

K l1 =1

δ2

[δ log(δ) +

(α− r)2

2σ2+ r − 1

]

46

Hence the solution to the optimal control problem in this case is given by

ϕ(s,w) = e−δs

(1

δlogw +

1

δ2

[δ log(δ) +

(α− r)2

2σ2+ r − 1

])(3–21)

This solution is only valid if the wealth process is strictly positive i.e. Wt > 0 for all time

t, in particular we must have that the initial wealth satisfies w > 0. Using the same

parameters as the power utility case and plotting the value function over the same range

as before, we end up taking on negative values of ϕ(s,w). This can be seen in Figure

3− 3 below, this is a result of the models dependence on δ. In the power utility case

the parameters were restricted in such a way that the condition in Equation (3–12) is

satisfied. In the log utility case we may choose any delta we like so long as δ > 0. The

choice of δ = 1 cancels one of the log terms in Equation (3–21), which leads to the value

function taking on negative values on the given range. The condition on δ to allow the

value function to only take positive values is difficult if not impossible to find explicitly

due to the δ log(δ) term in the value function. This term restricts our ability to solve for

δ explicitly (could do more here). Using an iterative numerical argument we find that for

δ bigger than a value between 1.7 and 1.8 we have positive value function. Given the

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

0

1

2

3

4

Infinitemax= 4.310295min= −0.93955

Figure 3-3. Value function for log utility with no jumps with parameters,α = .16, r = .06, δ = 1, γ = .5, σ = .3

47

solution ϕ(s,w) as above we find the optimal controls

c∗ =w

K l0= δw θ∗ =

α− rσ2

(3–22)

Hence we have that ct = δWt , which means that we again have that the consumption

process is linear function of the wealth process. Also since δ > 0 we have the same

positively correlated relationship between the wealth and consumption process as in the

power law case. That is, an increase in the wealth of the investor leads to an increase

in the investors consumption, which is a result that matches real world intuition. The

fraction of wealth in the risky asset is given by θt =α−rσ2

which is again constant in time.

To complete the analysis of this problem we use that fact that

ct = δWt θt =α− rσ2

(3–23)

to show that the wealth process becomes

dWt = (α− r)θtWt dt + [rWt − ct ]dt + σθtWt dBt

=(α− r)2

σ2Wt dt + rWt dt − δWt dt +

α− rσWt dBt

=

[(α− r)2

σ2+ r − δ

]Wt dt +

α− rσWt dBt

hence we have the wealth process is a Geometric Brownian motion of the form

dWtWt=

[(α− r)2

σ2+ r − δ

]dt +

α− rσdBt Ws = w (3–24)

whose solution exists and unique given the initial conditionWs = w . The explicit solution

to this Geometric Brownian motion is given by

Wt = w e

([(α−r)2

σ2+r−δ

]− 12(α−r)2

σ2

)(t−s)+α−r

σ

√t−sN(0,1)

(3–25)

where N(0, 1) is a standard normal random variable. A simulation of one path of the

wealth process using the same parameters as before is presented in Figure 3− 4. To

show that the solution ϕ(s,w) is optimal we need to show that the conditions of the

48

Total Wealth for Log Utility

Time

W_t

0.0 0.2 0.4 0.6 0.8 1.0

0.80

0.90

1.00

1.10

Figure 3-4. Wealth process for Log utility with no jumps with parameters,w = 1,α = .16, r = .06, δ = 1, σ = .3

verification theorem are satisfied so that the methodology is legitimate for this problem.

We first need to show that the control process ut = (ct , θt) is an element of A[s,∞). The

precise form of the control process is

ut =

(δWt ,

α− rσ2

)which is measurable and Ft-adapted. We next check that the control process satisfies

the following integrability condition

E s,w[∫ ∞

s

||ut ||2 dt]= E s,w

[∫ ∞

s

δWt)

2 +

(α− rσ2

)2dt

]

= δ2E s,w[∫ ∞

s

W 2t dt

]+

(α− rσ2

)2Ps,w [s,∞) < ∞

The first term is finite sinceWt is a unique strong solution, and the second terms is finite

since we are working on a probability space. The last condition for admissibility is to

check the finiteness of the cost functional using the control we have found,

Jθ,c(s,w) = E s,w[∫ ∞

s

e−δt log(δWt) dt

]

49

≤ E s,w[∫ ∞

s

(log δ + logWt) dt

]≤ E s,w

[∫ ∞

s

log δ dt

]+ E s,w

[∫ ∞

s

(1 +W 2t ) dt

]≤

((1 + log δ)Ps,w [s,∞) + E s,w

[∫ ∞

s

W 2t dt

])< ∞

Hence the control process ut ∈ A[s,∞), to finish the verification of the technical

conditions we must verify that the hypotheses of the verification theorem are satisfied.

First we note that ϕ(s,w) ∈ C 1,20 , is continuous, and satisfies the Hamilton Jacobi

Bellman equation by construction of the constant δ. The growth condition also holds

since

|ϕ(s,w)| = |eδsδ logw | ≤ δ| logw | ≤ δ(1 + |w |2) (3–26)

A similar calculation shows that the conditions for f u(s,w) = e−δs log c and σu(s,w) =

σθw also hold, so that the candidate ϕ(s,w) is actually the optimal solution to the

problem i.e. ϕ(s,w) = Φ(s,w). Although the logarithmic utility provides a closed form

solution in this setting, it will turn out that a closed form solution in the finite horizon case

cannot be found using the methodology used in this paper. We will provide the closed

form solution in the case with jumps for an infinite horizon in a later section, and we will

show why a closed form solution is not tractable in a later section. Before moving on the

the cases with jumps we consider the power utility case over the finite horizon [0,T ].

Once finding a closed form solution the the optimal control problem we show that in the

limit as T → ∞ the solution we find will converge to the solution for the infinite horizon.

This will be shown explicitly in the formula and also using numerical arguments, to show

this we proceed in a similar manner as before.

3.4 Finite Horizon Power Utility

In this section we would like to consider the a more general result of that which

we have already solved. In particular, we would like to consider the case of power

utility over a finite horizon [0,T ]. The finite horizon case does not allow us to use the

50

calculation which showed that Φ(s,w) = e−δsΦ(0,w) since we do not retain the same

form after a substitution. The cost functional has the following form

Jθ,c(s,w) = E s,w[∫ Ts

e−δt cγt

γdt

]= E y

[∫ T−s

0

e−δ(s+t) cγs+t

γdt

](3–27)

We see that in our solution we will be able to factor out e−δs as before, however the

expectation is no longer just a function of w . The expectation will now involve both w

and the new stopping time T − s, we will explicitly find this expectation and show that

this solution generalizes the case when T → ∞. The Hamilton Jacobi Bellman equation

for this problem is the same as the infinite horizon case and is given by Equation (3–9),

hence the optimal controls are given by Equation (3–10). In the finite horizon case the

Hamilton Jacobi Bellman theorem adds an initial condition from the fact that Φ(y) = g(y)

for all y ∈ ∂G . For this particular problem this equation translates to ϕ(T ,w) = 0,

plugging the optimal controls into the Hamilton Jacobi Bellman equation as before we

end up with the partial differential equation with boundary condition given by

1− γ

γ(eδs)

1γ−1ϕ

γγ−1w + ϕs − (α− r)2ϕ2w

2σ2ϕww+ rwϕw = 0

ϕ(T ,w) = 0

To find a solution we may use the idea of separation of variables, the function of w will

retain the same form as the infinite case but the time component will be different as it

now involves T − s. Assuming the solution has the form ϕ(s,w) = f (s)w γ we may take

the necessary derivatives and plug into the Hamilton Jacobi Bellman equation to get

1− γ

γ(eδs)

1γ−1[f (s)γw γ−1] γ

γ−1 + f ′(s)w γ − (α− r)2f 2(s)γ2w 2γ−2

2σ2f (s)γ(γ − 1)w γ−2 + rwf (s)γwγ−1 = 0

Simplifying and dividing out by w γ we end up with the following Bernoulli ordinary

differential equation

f ′(s) +

[rγ − (α− r)2γ

2σ2(γ − 1)

]f (s) = (γ − 1)(γeδs)

1γ−1 f (s)

γγ−1 (3–28)

51

f (T ) = 0 (3–29)

To recognize this equation as a Bernoulli equation and then write down the general

solution let η =: rγ − (α−r)2γ2σ2(γ−1) = P(s), q(s) = (γ − 1)(γeδs)

1γ−1 and n = γ

γ−1 then Equation

(3–28) becomes

f ′(s) + p(s)f (s) = q(s)f n(s)

The general solution of this type of equation is given by

f (s) =

[(1− n)

∫e(1−n)

∫p(s) dsq(s) ds + C

e(1−n)∫p(s) ds

] 11−n

(3–30)

as long as n = 1. Using this general solution we find that the solution for our problem is

given by

f (s) =

[−(γ)

1γ−1

βe

δγ−1 s + Ce

ηγ−1 s

]1−γ

where β = δ−ηγ−1 . Using the boundary condition f (T ) = 0 we find that the constant is

C = (γ)1

γ−1

βe

δ−ηγ−1T which leads to the general solution

f (s) =

[−(γ)

1γ−1

βe

δγ−1 s +

(γ)1

γ−1

βe

δγ−1T

]1−γ

=

[(γ)

1γ−1 e

δγ−1 s

β

(−1 + e

δγ−1 (T−s)

)]1−γ

=βγ−1

γe−δs

[−1 + e

δγ−1 (T−s)

]1−γ

=1

γ

[1

1− γ

(δ − γr − (α− r)2γ

2σ2(1− γ)

)]γ−1e−δs

[1− e

δγ−1 (T−s)

]1−γ

Hence we have that the value function in this case is given by

ϕ(s,w) =1

γ

[1

1− γ

(δ − γr − (α− r)2γ

2σ2(1− γ)

)]γ−1e−δs

[1− e

δγ−1 (T−s)

]1−γ

w γ (3–31)

52

As in the infinite horizon case is only valid if

δ > γ

[r +

(α− r)2

2σ2(1− γ)

]so that we avoid the value function being being infinite. Letting T → ∞ in Equation

(3–31) since T − s ≥ 0 and δγ−1 < 0 we see that the term e

δγ−1 (T−s) → 0 and we

get the solution for the infinite horizon case. This can serve as a check of theoretical

consistency for the formula we have derived in the finite horizon case. I have also used

the fact that the finite horizon case should recover the infinite horizon case as T → ∞

to check my numerical results and the validity of the programs I have written. The first

of these checks of consistency comes from the plots of the value function for increasing

values of T . As T gets large we should have that the plots of the value function in

the finite horizon case should approach that of the infinite horizon case, the results

of the plots for T = 1, 2, 100 are presented in Figure 5− 1. For T = 1 we clearly

see a difference between the plots, however as soon as T = 2 it becomes difficult to

differentiate that plot from the infinite horizon case. To show there is in fact a difference

between the plots I have provided the maximum and minimum values of ϕ(s,w) for

comparison. For the values of T = 1 and T = 2 we see the max/min values differ,

implying the graphs and values of ϕ(s,w) are different. Once T = 100 (actually sooner)

we see that the values for that case and the infinite horizon case are the same, which

means the plots are the same and hence the models are indistinguishable for all intents

and purposes. We also present more numerical validation of the finite horizon model

by considering the maximum differences between the model for each finite time and the

infinite time horizon. We expect that as T gets larger the differences should tend to zero,

since the term (1 − eδ

γ−1 (T−s)) tends to one and we recover the infinite horizon case.

Our expected results are confirmed in Table 3− 1, again once T = 100 we see that

the maximum difference of this case with the infinite horizon case is zero which means

the values of ϕ(s,w) are the same in both cases. To find the consumption process we

53

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

0

1

2

3

4

T= 1max= 4.222339

max= 0

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

T= 2max= 4.498992

max= 0.4912003

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

T= 100max= 4.540767max= 0.5282442

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4


Figure 3-5. Plots of the value function for values of T = 1, 2, 100 compared to the plot ofthe value function for infinite horizon

Table 3-1. Maximum differences between finite horizon and infinite horizon for differentvalues of T

T=1 T=2 T=3 T=5 T=100Max difference 1.6704549229 0.1171431654 0.0153684204 0.0002802111 0.0000000000

rewrite Equation (3–31) in a simpler form so that we may take the derivative, combining

the terms in the brackets we get Equation (3–32). Upon closer inspection we see that

Equation (3–32) is almost the same as Equation (3–11), the only difference comes from

the term (1− eδ

γ−1 (T−s)). Intuitively this says that the finite horizon case is essentially the

infinite horizon case discounted by an appropriate factor.

ϕ(s,w) =1

γ

δ − η

(1− γ)(1− e

δγ−1 (T−s)

)γ−1

e−δsw γ (3–32)

54

Given the compact form in Equation (3–32) we may calculate ϕw and plug into the

equation for c∗ = (e−δsϕw)1

γ−1 to find that

c∗ =

δ − η

(1− γ)(1− e

δγ−1 (T−s)

)w (3–33)

Hence for general time we have that the optimal controls are

ct =

δ − η

(1− γ)(1− e

δγ−1T

)Wt θt =

(α− r)σ2(1− γ)

Plugging these optimal controls into Equation (3–5) we find that the wealth process is a

Geometric Brownian motion of the form

dWtWt=

(α− r)2

σ2(1− γ)+ r − δ − η

(1− γ)(1− e

δγ−1T

) dt + α− r

σ(1− γ)dBt Ws = w

Checking the conditions of the verification theorem in the finite horizon case is slightly

more difficult than the infinite horizon case. The added complication comes from the fact

the function ϕ(s,w) = f (s)w γ so that we cannot just pull out the constant when checking

the boundedness conditions. We first show that the control process is an element of

A[s,T ], the control process in this case is given by

ut =

δ − η

(1− γ)(1− e

δγ−1T

)Wt , (α− r)

σ2(1− γ)

It turns out that the constant in brackets is positive, we already know that δ > η and

1 − γ > 0. The only term that needs checking is the term involving the exponential. If

we rewrite this term as 1 − e−δ1−γT , then for all T ≥ 0 we have a negative exponential

function hence we have that e−δ1−γT is a decreasing function for T ∈ [0,∞) so the

maximum occurs at T = 0 i.e. e−δ1−γT ≤ 1 as needed. Following the same argument as

55

before we can show that

E s,w[∫ Ts

||ut ||2 dt]=

δ − η

(1− γ)(1− e

δγ−1T

)2 E s,w [∫ T

s

W 2t dt

](3–34)

+

(α− r

σ2(1− γ)

)2Ps,w [s,T ] < ∞

We next check the finiteness of the cost functional using the control process we have

found for this problem. This cost functional only differs from the infinite case in that the

integral is over a finite interval and the constant being multiplied by the wealth process

is different. Hence this gives that the argument is essentially the same with only a slight

modification, I present it for completeness

Jθ,c(s,w) = E s,w

1γ

∫ Ts

e−δt

δ − η

(1− γ)(1− e

δγ−1T

)Wtγ

dt

≤ 1

γ

δ − η

(1− γ)(1− e

δγ−1T

)γ

E s,w[∫ Ts

W γt dt

]

≤ 1

γ

δ − η

(1− γ)(1− e

δγ−1T

)γ

E s,w[∫ Ts

(1 +W 2t ) dt

]

≤ 1

γ

δ − η

(1− γ)(1− e

δγ−1T

)γ (

Ps,w [s,T ) + E s,w[∫ Ts

W 2t dt

])< ∞

Since the control process ut is Ft adapted we have that ut ∈ A[s,T ], given this

admissible strategy we need only check the remaining conditions of the verification

theorem. The function ϕ ∈ C 1,20 is continuous and satisfies the conditions of the Hamilton

Jacobi Bellman equation. To show that the growth condition is satisfied we first need to

perform some elementary analysis on the exponential function. Consider the function

f (x) = e−δx1−γ , then the derivative f ′(x) = − δ

1−γe−

δx1−γ < 0 for all x hence the function f is

decreasing. In particular since T − s ≤ T for all s ∈ [0,∞) we have that

e−δ1−γ(T−s) ≥ e−

δ1−γT

56

hence we have the following relation that will help us find a bound for the term in

parentheses

1− eδ

γ−1 (T−s) ≤ 1− eδ

γ−1T

With this preliminary result and writing a positive power in the exponent we may now

show that the candidate for the value function is bounded

|ϕ(s,w)| =

∣∣∣∣∣∣∣1

γ

(1− γ)(1− e

δγ−1 (T−s)

)δ − η

1−γ

e−δsw γ

∣∣∣∣∣∣∣≤ 1

γ

(1− γ)(1− e

δγ−1T

)δ − η

1−γ

|w |γ

≤ 1

γ

(1− γ)(1− e

δγ−1T

)δ − η

1−γ

(1 + |w |2)

The remaining conditions of the verification theorem have already been verified hence

we have that the candidate for the solution ϕ(s,w) is actually the solution of the optimal

control problem i.e. ϕ(s,w) = Φ(s,w). We note that in any of the equations above a

check of consistency between the finite horizon case and the infinite horizon case by

letting T → ∞ in any of the above equations gives the correct equation. Hence we

have generalized the infinite horizon case of the hyperbolic absolute risk aversion utility

function U(c) = cγ

γso that the finite horizon case includes both cases. An attempt

at generalizing the logarithmic case was made but a result was not found due to an

equation arising which does not have an analytic solution. Hence a closed form result

was not possible as in the case of the logarithmic function, however one can easily use

numerical methods to find the solution of the non analytic equation which arises. Before

moving on to the case of Levy processes with jumps we provide the attempted analysis

for the logarithmic case for completeness.

57

3.5 Finite Horizon Log Utility

In this section we explore the case of logarithmic utility over a finite time horizon, it

turns out that my attempt at a solution in this case was unsuccessful. The reason for this

is that we end up with an equation which has no analytic solution, however I provide the

attempt for completeness. Since we are in the finite case we are not able to completely

remove the time dependence from the value function. To see this we consider the cost

functional of the form


e−δt log ct dt

]= E y

[∫ T−s

0

e−δ(s+t) log cs+t dt

](3–35)

We see that even after factoring out the e−δs term the remaining expectation still involves

the time parameter, in particular we have that ϕ(s,w) = e−δs ϕ(s,w). The Hamilton

Jacobi Bellman equation with boundary condition for this problem is the same as the

infinite horizon case

supθ,c


1

2σ2w 2θ2ϕww

]= 0 ϕ(T ,w) = 0

Using the fact that we may remove the e−δs from the value function we may rewrite the

Hamilton Jacobi Bellman equation in the following simplified form

supθ,c

[log c − δϕ+ ϕw + θ(α− r)w ϕw + (rw − c)ϕw +

1

2σ2w 2θ2ϕww

]= 0 ϕ(T ,w) = 0

Assuming the value function ϕ has the form ϕ(s,w) = g(s) logw , taking necessary

derivatives and plugging into the Hamilton Jacobi Bellman we get

supθ,c

[log c − δg(s) logw + g′(s) logw + θ(α− r)g(s) + rg(s)− c

wg(s)− 1

2σ2θ2g(s)

]= 0

The boundary condition now states that ϕ(T ,w) = e−δTg(T ) logw = 0, we note

that there are two possible cases for this product to be zero. Either w = 1 and g(T )

is allowed to be any real number or w = 1 and g(T ) = 0. In either case the optimal

58

controls are

c∗ =w

g(s)θ∗ =

(α− r)σ2

With these optimal controls the Hamilton Jacobi Bellman equation becomes the

following ordinary differential equation[log

(w

g(s)

)− δg(s) logw + g′(s) logw +

(α− r)2

σ2g(s) + rg(s)− 1− (α− r)2

2σ2g(s)

]= 0

Simplifying this expression and combining like terms we end up with the following

equation

[1− δg(s) + g′(s)] logw +

[(α− r)2

2σ2+ r

]g(s)− log(g(s))− 1 = 0 (3–36)

In either case we will have the problem of not being able to solve this problem

analytically, however we show that both cases reduce to solving the same equation.

• case 1: We first consider the case where w = 1 then from Equation (3–36) thecoefficient of logw must be equal to zero which gives the following equation withboundary condition

g′(s)− δg(s) = −1 g(T ) = 0

This is a linear ODE whose solution is given by

g(s) =1

δ+ Ceδs

The problem of solving this equation is that we need to satisfy Equation (3–36) sowe must also have that the remaining constant term which does not depend on wmust also be equal to zero i.e.[

(α− r)2

2σ2+ r

]g(s)− log(g(s)) = 1

• case 2: This case essentially has the same problem as the last case, if w = 1 thelog term from Equation (3–36) disappears and we end up with the same equationas before [

(α− r)2

2σ2+ r

]g(s)− log(g(s)) = 1 g(T ) ∈ R

59

In either case we end up with an equation which is not able to be solved analytically

so the methodology I used in the infinite horizon case is not able to provide a solution.

However there may exists another methodology which may be able to provide a result

which generalizes the infinite horizon case. The same difficulty arises in later sections

when we assume the underlying stock price follows a Levy process so I will not present

that case explicitly. If we were able to find a solution in this case the generalization to the

jump case would follow the same methodology that will be used in future sections.

60

CHAPTER 4LEVY PROCESS

4.1 Why use a Levy Process with Jumps?

In the cases above we assume the risky asset follows a geometric Brownian motion,

so that the asset returns are log-normally distributed. The assumption of asset prices

following a Geometric Brownian motion has several problems when compare to real

world phenomenon. The first of these is the fact that Geometric Brownian motion

assumes that stock prices follow a continuous path, however many equity prices exhibit

large corrections which behave like discontinuous jumps in the price. Much of the risk

associated with equity price movements occurs during the periods of the correction, so

in order to accurately model the prices we may require jump diffusion models. Further

evidence that suggests Jump diffusion models may be useful in modeling the behavior of

stock prices comes from considering the returns on asset prices. Given the assumption

of Geometric Brownian motion we have that log of asset returns are normally distributed,

this does not match the empirically collected data from real world data. Many asset

prices exhibit a distribution of returns which have so called ”heavy tails”. This means

that the distributions decay slowly to zero at the extremes, the result of this is that rare

(tail) events are assigned larger probabilities of occurring than a normal distribution. This

leads to the model underestimating the risk of large movements in the asset prices that

is typically observed in a real world setting. Hence a more accurate model of real world

asset prices should exhibit higher probability of tail events than the Brownian motion

model. The library of available diffusion models is very extensive, and there are some

diffusion models which may be able to replicate a heavy tailed distribution, for example

we may use stochastic volatility models to allow for greater variance in the asset prices

leading higher probability of tail events. However, no matter how accurately we are able

to model returns of asset prices, it is much more difficult to use Brownian motion to

model sudden jumps in the prices on the time scale we are interested in.

61

4.2 Preliminaries of Levy Process

We would now like to consider the optimal control problem when the stock price

is driven by a more general process than geometric Brownian motion. In order to do

this we need to study a process whose underlying distribution may not have a normal

distribution. As in the case of a process driven by Brownian motion, this new process

should have independent and stationary increments. Hence we will use essentially the

same assumptions for our new process with the major difference lying in the fact that

increments are no longer normally distributed. These so called Levy processes may

have a much more general underlying distribution which include the normal distribution

case as before. The study of Levy processes is extensive, and we will only provide

a short introduction of the material necessary to solve our problem. An extensive

presentation of Levy processes and the theoretical framework can be found in books

such as [1], [4], [25]. The application of Levy processes to the financial markets is still at

an elementary stage and more applied descriptions of Levy process in finance can be

found in [8] and [26].

Definition 4.1. Let (Ω,F , Ftt≥0,P) be a filtered probability space. An Ft adapted

stochastic process (Xt)t≥0 is called a Levy process if the following conditions hold

(i) X0 = 0 a.s

(ii) Given a partition t0 < t1 < · · · < tn the random variables Xt1 − Xt0, · · · ,Xtn − Xtn−1are independent. (Independent increments)

(iii) Xt+h − Xt has the same distribution as Xh − X0 for all t and h (Stationarity).

(iv) For every ϵ > 0 and h ≥ 0

limt→hP(|Xt − Xh| > ϵ) = 0

Just as in the case of stock prices driven by Brownian motion we must extend our idea

of Newtonian Calculus to include these more general Levy processes. The extension is

aided by what we already now about the Ito Calculus, but must be extended to include

62

jump processes which are able to be modeled in the Levy process case. In order to

perform analysis on a stock price driven by a Levy process we must define integrals with

respect to a Levy process, the main extension will be due to integrals with respect to a

discontinuous process. To get to the point of defining integrals against a discontinuous

measure we need the following background on jump processes.

Definition 4.2. Let (Xt)t≥0 be a Levy process, we define the jump process by the

process (∆Xt)t≥0 where ∆Xt = Xt − Xt−

If we consider the jumps of a Levy process directly we find that a difficulty arises from

the fact that we may have∑0≤s≤t |∆Xs | = ∞ a.s. To avoid this problem we would like to

consider jumps of a specified size so that we may have a σ-finite measure to integrate

with respect to. The following definition allows us to avoid this difficulty.

Definition 4.3. Let t ∈ [0,∞) and U ∈ B(Rd − 0) we define the Poisson random

measure

N(t,U)(ω) = #0 ≤ s ≤ t; ∆Xs(ω) ∈ U =∑0≤s≤t

χU(∆Xs(ω))

If we fix t ≥ 0 and ω ∈ Ω we see that N(t, ·) is a counting measure on B(Rd − 0)

which counts the number of jumps on the interval [0, t] of size ∆Xs in a given Borel set

U. Alternatively if we fix t and U we have that N(t,U) is a random variable defined on

Ω so that the quantity E [N(t,U)] =∫N(t,U)(ω)dP(ω) is well defined, this leads to the

following definition

Definition 4.4. Let (Xt)t≥0 be a Levy process and N(t, ·) be the associated Poisson

random measure then

ν(·) = E [N(t, ·)]

is a Borel measure on B(Rd − 0) called the Levy measure.

It can be shown that if 0 /∈ U then N(t,U) < ∞ for all t ≥ 0, so that the map ω 7→

N(t,U)ω is a σ-finite measure, and implication of this result is that ν(U) < ∞ so that ν is

63

a σ-finite measure also. As a random variable the Poisson random measure N(t,U) has

a particularly familiar form that leads to many useful properties.

Theorem 4.1. For every U ∈ B(R) with 0 /∈ U we have that (N(t,U))t≥0 is a Poisson

process with intensity λ = ν(U), in particular

P[N(t,U) = n] =e−λt(λt)n

n!n = 1, 2, 3, · · ·

Proof. A proof of can be found in [1] or other introductory books on Levy process.

This property is useful to keep in mind when performing calculations involving Poisson

integrals, a particularly useful result is that integrals with respect to this Poisson random

measure will turn out to be a compound Poisson process. To see this and the other

useful properties we provide a formal definition of the integral with respect to N(t,U),

Definition 4.5. Let f be a Borel measurable function on U with 0 /∈ U we define the

Poisson integral of f to be∫U

f (z)N(t, dz) =∑z∈U

f (z)N(t, z) =∑0<u≤t

f (∆X (u))χU(∆X (u))

The study of Poisson integrals is not the main focus of this paper so we will not provide

the intricate details of the theory, much of this theory may be found in [1], [4] or [25].

However there are some useful properties that provide some insight into the theory for

those familiar with integration with respect to Brownian motion. Much of the theory of

Brownian integrals can be extended to the case of Poisson integrals and this is done

precisely in any of the aforementioned books. We provide just a summary of some the

useful properties that will help guide us through the remainder of the paper

Theorem 4.2. (Properties of Poisson integrals) Let f be a Borel measurable function

of U with 0 /∈ U and N(t,U) the Poisson random measure associated with a Poisson

process (Xt)t≥0 then the random variable

Zt =:

∫U

f (z)N(t, dz)

64

satisfies the following properties

• For all t ≥ 0, the random variable Zt has a compound Poisson distribution withcharacteristic function

ϕZt(u) = E [eiuZt ] = et

∫U(e iuz−1)νf (dz) for all u ∈ R

where νf = ν f −1.

• If f ∈ L1(U, ν|U) then

E [Zt ] = t

∫U

f (z)ν(dz)

• If f ∈ L2(U, ν|U) then

Var [Zt ] = t

∫U

|f (z)|2ν(dz)

• (Zt)t≥0 is a compound Poisson process.

• (Zt)t≥0 is a martingale, where

Zt = Zt − t∫U

f (z)ν(dz)

The most useful of the these properties is the fact that Zt is a martingale, this gives

that for a ”reasonable” function f (z) the expectation of Zt will be equal to zero. For our

analysis we will be considering the particularly simple function f (z) = z , whose integral

with respect to Poisson measure has the following useful interpretation. The Poisson

integral ∫U

z N(t, dz) =∑0<u≤t

∆X (u)χU(∆X (u))

is the sum of all jumps (more specifically the sum of all jump sizes) taking values in the

set U up to time t. It turns out that counting the size of all the jumps (i.e. f (z) = z) is

all that is necessary to classify the set of all Levy processes as the following theorem

shows.

65

Theorem 4.3. (Levy-Ito Decomposition) Let (Xt)t≥0 be a Levy process, then Xt has the

decomposition

Xt = αt + σBt +

∫ t0

∫|z |<R

z N(ds, dz) +

∫ t0

∫|z |≥R

z N(ds, dz) (4–1)

for constants α,σ ∈ R, R ∈ [0,∞] where

N(ds, dz) = N(ds, dz)− ν(dz)ds

is the compensated Poison random measure of X(·) and Bt is an independent Brownian

motion.

Proof. A proof by construction can be found in Applebaum [1].

Theorem 4.3 gives the decomposition for any general Levy process, however we will not

need the full generality it provides. In particular if we consider the set of Levy process

with finite expectation we can choose R = ∞ in Equation (4–1), then we get a simplified

version of the general Levy-Ito process

Theorem 4.4. If E |Xt | < ∞ for all t ≥ 0 in Equation (4–1) then we may choose R = ∞

and hence we may write

Xt = αt + σBt +

∫ t0

∫RzN(ds, dz)

Proof. The proof of this depends on theorem 25.3 of Sato [25], which states that

E |Xt | < ∞ if and only if∫|z |≥R z ν(dz) < ∞. This implies that the integral∫ t

0

∫|z |≥R

z ν(dz) ds < ∞

hence we may add and subtract this integral to Equation (4–1) to get

Xt = αt + σBt +

∫ t0

∫|z |<R

z N(ds, dz) +

∫ t0

∫|z |≥R

z N(ds, dz)

−∫ t0

∫|z |≥R

z ν(dz) ds + t

∫|z |≥R

z ν(dz)

66

=

(α+

∫|z |≥R

zν(dz)

)t + σBt +

∫ t0

∫|z |<R

z N(ds, dz) +

∫ t0

∫|z |≥R

z N(ds, dz)

= α t + σBt +

∫ t0

∫Rz N(ds, dz)

The Levy-Ito decomposition gives us a way of writing every Levy process as a sum

of a continuous part plus a discontinuous part, however it does not tell give us any

information about the existence of Levy processes. The matter of existence of Levy

process is due to the following theorem, which gives conditions for existence and a

general formula for the characteristic function of a Levy process.

Theorem 4.5. (Levy-Khintchine theorem) Given a Levy process (Xt)t≥0 with Levy

measure ν, then∫Rmin(1, z

2)ν(dz) < ∞ and

E [e iuXt ] = etη(u), u ∈ R

where the quantity η(u) called the Levy exponent is given by

η(u) = −12σ2u2 + iαu +

∫|z |<R(e iuz − 1− iuz)ν(dz) +

∫|z |≥R(e iuz − 1)ν(dz)

Conversely, given constants α,σ2 and a measure ν on R such that∫Rmin(1, z2)ν(dz) < ∞

there exists a unique Levy process Xt such that the characteristic function is given as

above.

Proof. A proof by construction and a proof by characteristic funcitons can be found in

Applebaum [1].

4.3 Ito-Levy Diffusion and it’s Generator

Throughout this section as with the rest of the paper we will be working with one

dimensional processes so all theorems will be stated for this case only, however the

interested reader may consult [1] or [21] for the multidimensional cases.

67

Theorem 4.6. Consider the Ito-Levy process stochastic differential equation on [s,T ] of

the form

dXt =

[α(t,w)−

∫|z |<R

γ(t, z ,w)ν(dz)

]dt+σ(t,w)dBt+

∫Rγ(t,w , z)N(dt, dz) Xs = x

where α : [s,T ] × R → R, σ : [s,T ] × R → R and γ : [s,T ] × R × R → R satisfy the

following Lipschitz and at most linear growth conditions

• There exists constant C1 > 0 such that for all x , y ∈ R

|α(t, x)− α(t, y)|2 + |σ(t, x)− σ(t, y)|2

+

∫R|γ(t, x , z)− γ(t, y , z)|2 ν(dz) ≤ C1|x − y |2

• There exists constant C2 > 0 such that for all x ∈ R

|α(t, x)|2 + |σ(t, x)|2 +∫R|γ(t, x , z)|2 ν(dz) ≤ C2(1 + |x |2)

Then there exists a unique cadlag adapted solution Xt such that

E s,x[∫ Ts

|Xt |2]< ∞

Throughout the paper we will be dealing with the so called Geometric Levy process

which satisfies the conditions of the for the existence and uniqueness for a solution of

the corresponding stochastic differential equation. The general form of a Geometric Levy

process in differential form is given by

dXt = αXt dt + σXt dBt + Xt−

∫Rγ(t, z)N(dt, dz) Xs = x (4–2)

whose unique solution can be found by applying the Ito-Levy theorem to ln(Xt). The

Ito-Levy theorem is an extension of Ito’s lemma to include processes with jumps, hence

it will be useful to consider a Levy process as a sum of continuous and discontinuous

parts. The Levy-Ito decomposition 4.3 allows us to break every Levy process into two

pieces, once this is done we may apply the classical Ito’s lemma to the continuous

part. Once we have found the differential for the continuous part one may apply an

68

argument to figure out how the discontinuous part changes before and after jumps. This

interested reader can find this argument in [21], we provide the result of that analysis in

the Ito-Levy theorem

Theorem 4.7. (One dimensional Ito-Levy theorem)Suppose Xt ∈ R is an Ito-Levy

process of the form

dXt =

[α(t,w)−

∫|z |<R

γ(t, z ,w)ν(dz)

]dt+σ(t,w)dBt+

∫Rγ(t,w , z)N(dt, dz) Xs = x

for some R ∈ [0,∞]. Let ϕ ∈ C 1,20 ([0,∞,R) and define Yt = ϕ(t,Xt), then Yt is an

Ito-Levy process and

dYt =∂ϕ

∂t(t,Xt)dt +

∂ϕ

∂x(t,Xt)dX

ct +1

2

∂2ϕ

∂x2(t,Xt)dX

ct dX

ct

+

∫R[ϕ(t,Xt− + γ(t,Xt−, z))− ϕ(t,Xt−)]N(dt, dz) (4–3)

where

dX ct =:

[α(t,w)−

∫|z |<R

γ(t,w , z)ν(dz)

]dt + σ(t,w)dBt

Using the Ito-Levy theorem we may write down the solution of the Geometric Levy

process given by Equation (4–2)

Xt = x exp

[(α− 12σ2)t + σBt +

∫ t0

∫|z |<R

[ln(1 + γ(s, z))− γ(s, z)] ν(dz) ds

+

∫ t0

∫Rln(1 + γ(s, z))N(ds, dz)

]We note that this equation is consistent with the Geometric Brownian motion case when

N = 0 as it should be. Also note that we must have the restriction that 1 + γ(s, z) > 0 so

that the logarithmic term is well defined. For the Ito-Levy process we will be considering

we will have that γ(s, z) = z , this implies that we will need to consider jump sizes larger

than −1 so that this equation is well defined. To perform the analysis in the jump case

using diffusion theory we need to be able to use an equivalent version of the Hamilton

Jacobi Bellman theorem. In order to write down the Hamilton Jacobi Bellman equation

69

for the case with jumps we need to compute the infinitesimal generator of an Ito-Levy

process, more specifically we need the generator of an Ito-Levy diffusion. The general

version of the Ito-Levy process presented in Theorem 4.7 can be simplified by assuming

that E |Xt | < ∞ as before. Given this assumption we have that the most general form

reduces and Ito process of the form found in Theorem 4.4, this will be the version we

use throughout the paper. The infinitesimal generator for a Levy process with jumps is

found by using the Ito-Levy theorem with the fact that N is a martingale, the statement

and proof are provided in the following theorem.

Theorem 4.8. Let Xt be an Ito-Levy diffusion of the form

dXt = α(t,Xt)dt + σ(t,Xt)dBt +

∫Rγ(t,Xt−, z)N(dt, dz) Xs = x

if ϕ ∈ C 1,20 ([0,∞),R) then Aϕ(s, x) exists for all x ∈ R and s ∈ [0,∞) and is given by

Aϕ(s, x) =∂ϕ

∂t(s, x) + α(s, x)

∂ϕ

∂x(s, x) +

1

2σ(s, x)

∂2ϕ

∂x2(s, x)

+

∫R

[ϕ(s, x + γ(s, x , z))− ϕ(s, x)− ∂ϕ

∂xγ(s, x , z)

]ν(dz) (4–4)

Proof. Since ϕ ∈ C 1,20 ([0,∞),R) we may apply Ito’s formula to compute

dϕ(t,Xt) =∂ϕ

∂t(t,Xt) +

∂ϕ

∂x(t,Xt)dX

ct +1

2

∂2ϕ

∂x2(t,Xt)dX

ct dX

ct

+

∫R[ϕ(t,Xt− + γ(t,Xt−, z))− ϕ(t,Xt−)]N(dt, dz)

=

[∂ϕ

∂t(t,Xt) + α(t,Xt)

∂ϕ

∂x(t,Xt) +

1

2σ2(t,Xt)

∂2ϕ

∂x2(t,Xt)

]dt

+ σ(t,Xt)∂ϕ

∂x(t,Xt) dBt +

∫R[ϕ(t,Xt− + γ(t,Xt−, z))− ϕ(t,Xt−)]N(dt, dz)

−∫R

∂ϕ

∂x(t,Xt)γ(t,Xt−, z)ν(dz) dt

If we now add and subtract the term∫R[ϕ(t,Xt− + γ(t,Xt−, z))− ϕ(t,Xt−)] ν(dz) dt

70

from the right had side of the equation for dϕ(t,Xt) we end up with a martingale term

involving N(dt, dz) which we may eliminate when we take expectations. Before taking

expectations we have that

dϕ(t,Xt) =

[∂ϕ

∂t(t,Xt) + α(t,Xt)

∂ϕ

∂x(t,Xt) +

1

2σ2(t,Xt)

∂2ϕ

∂x2(t,Xt)

]dt

+

∫R

[ϕ(t,Xt− + γ(t,Xt−, z))− ϕ(t,Xt−)−

∂ϕ

∂x(t,Xt)γ(t,Xt−, z)

]ν(dz) dt

+

∫R[ϕ(t,Xt− + γ(t,Xt−, z))− ϕ(t,Xt−)] N(dt, dz) + σ(t,Xt)

∂ϕ

∂x(t,Xt) dBt

=: Aϕ(t,Xt) dt +

∫R[ϕ(t,Xt− + γ(t,Xt−, z))− ϕ(t,Xt−)] N(dt, dz)

+ σ(t,Xt)∂ϕ

∂x(t,Xt) dBt

Integration of both sides of the equation with respect to the proper measures we find

that

ϕ(t,Xt)− ϕ(s,Xs) =

∫ ts

Aϕ(r ,Xr) dr +

∫ ts

∫R[ϕ(r ,Xr− + γ(r ,Xr−, z))

−ϕ(r ,Xr−)] N(dr , dz) +

∫ ts

∂ϕ


using the fact that Xs = x and taking the expectation E s,x of both sides of the equation

we are able to compute the numerator of the limit


Aϕ(r ,Xr) dr

]+ E s,x

[∫ ts

∂ϕ


]+ E s,x

[∫ ts

∫R[ϕ(r ,Xr− + γ(r ,Xr−, z))− ϕ(r ,Xr−)] N(dr , dz)

]Given that the function ∂ϕ

∂x(r ,Xr)σ(r ,Xr) is B × F -measurable, Fr -adapted,

and E s,x[∫ ts

(∂ϕ∂x(r ,Xr)σ(r ,Xr)

)2dr]< ∞ a standard result from stochastic calculus

gives that

E s,x[∫ ts

∂ϕ


]= 0

71

Given that the measure N(dr , dz) is a martingale we have a similar integrability condition

for Poisson integrals

E s,x[∫ ts

∫R[ϕ(r ,Xr− + γ(r ,Xr−, z))− ϕ(r ,Xr−)]

2 ν(dz) dr

]< ∞

once this condition is satisfied we may use that fact that the martingale process is

constant on average which allows us to conclude that

E s,x[∫ ts


]= 0

hence we have that both martingale terms are removed when we take expectations so

that the equation for the infinitesimal generator simplifies to


Aϕ(r ,Xr) dr

]dividing both sides by t − s and taking limits gives

Aϕ(s, x) =: limt→s

E s,x [ϕ(t,Xt)]− ϕ(s, x)

t − s

= limt→sE s,x

[1

t − s

∫ ts

Aϕ(r ,Xr) dr

]= E s,x

[limt→s

1

t − s

∫ ts

Aϕ(r ,Xr) dr

]= E s,x

[d

dt

(∫ ts

Aϕ(r ,Xr) dr

)]= E s,x

[Aϕ(t,Xt)

]=

∂ϕ

∂t(s, x) + α(s, x)

∂ϕ

∂x(s, x) +

1

2σ(s, x)

∂2ϕ

∂x2(s, x)

+

∫R

[ϕ(s, x + γ(s, x , z))− ϕ(s, x)− ∂ϕ

∂xγ(s, x , z)

]ν(dz)

The argument is essentially the same as the case without jumps once we have applied

the Ito-Levy formula, for a different argument involving Fourier transforms the reader

may see theorem 3.3.3 of [1].

72

We may now a version of the Hamilton Jacobi Bellman equation for a Levy process with

jumps, the theorem is essentially the same as the case without jumps. The only major

difference comes from a change in the integrability condition, the remaining hypotheses

and the conclusion is the same as the case without jumps.

Theorem 4.9. Let ϕ ∈ C 2(G) ∩ C(G), suppose the following conditions hold

• Auϕ(y) + f u(y) ≤ 0 for all y ∈ G , u ∈ U

• ϕ−(Yτ)τ≤τG is uniformly integrable for all u ∈ A[0, τG ] and y ∈ G .

•

E y

[|ϕ(Yτ)|+

∫ τG

0

(|Auϕ(Yt)|+

∣∣∣∣σ(Yt)∂ϕ∂y (Yt)∣∣∣∣2

+

∫R|ϕ(Yt + γ(Yt , ut , z))− ϕ(Yt)|2ν(dz)

)dt

]< ∞

• YτG ∈ ∂S a.s. on χτS<∞ and limt→τS−

ϕ(Yt) = g(YτS )χτS<∞ a.s for all u ∈ A[0, τG ]

then ϕ(y) ≥ Ju(y) for all Markov controls u ∈ A and y ∈ G . Moreover if for all y ∈ G we

find a Markov control u = u0(y) such that

f u0(y)(y) + Au0(y)ϕ(y) = 0

then u∗t = u0(Yt) is optimal and ϕ(y) = Φ(y) = Ju∗(y).

Proof. The proof is the same as Theorem 2.2 the case with no jumps, the only

difference shows up in the generator Au which does not affect the argument from

Theorem 2.2. The extra integrability condition is to make sure the performance function

Ju(y) is finite, for the no jump case this was taken care of since the control u was

admissible.

4.4 Verification Theorem for Levy Process with Jumps

As before we will use the Hamilton Jacobi Bellman equation indirectly to solve the

optimal control problem, the solution will come from an equivalent verification theorem

for Levy jump processes. The statement and proof of the verification theorem in the

73

jump case is very similar to the non jump case in Theorem 2.3, so we will focus mainly

on the differences. The statement of the theorem only differs by the addition of a

condition for the function γ(r , x , z), and is given by

Theorem 4.10. Let u ∈ A[s, τG ] and (s, x) ∈ G and suppose the following conditions are

satisfied for all s ∈ [0, τG ] and x ∈ R

• ϕ ∈ C 1,2([0, τG)× R) is continuous on [0, τG ]× R and satisfies the quadratic growthcondition |ϕ(s, x)| ≤ Cϕ(1 + |x |2)

• ϕ satisfies the Hamilton Jacobi Bellman equation

supu∈A[s,τG ]

[f u(s, x) + Auϕ(s, x)] = 0 s ∈ [0, τG)

ϕ(τG , x) = g(τG , x)

• f u is continuous with |f u(s, x)| ≤ Cf (1 + |x |2 + ||u||2) for some constant Cf > 0.

• |σu(s, x)|2 ≤ Cσ(1 + |x |2 + ||u||2) for some constant Cσ > 0.

•∫R|γ(s, x , z)|2ν(dz) ≤ Cγ(1 + |x |2 + ||u||2)

then ϕ(s, x) ≥ Φ(s, x) for all (s, x) ∈ G . Moreover if uo(s, x) is the max of u 7→

f u(s, x) + Auϕ(s, x) and u∗ = u0(s,Xs) is admissible then ϕ(s, x) = Φ(s, x) for all

(s, x) ∈ G and u∗ is and optimal strategy i.e. Φ(s, x) = Ju∗(s, x).

Proof. For the proof we proceed as before until we find a difference, we then adjust

accordingly and refer the reader to the proof of Theorem 2.3 since the remaining

justification will be the same. Fix s ∈ [0, τG ] and x ∈ R, to get that the process X is

bounded we define the stopping time

τn = τG ∧ inft > s : |Xt − Xs | ≥ n

Let u ∈ A[s, τG ] be an admissible control and Xs = x then by the Ito-Levy theorem we

have that

ϕ(τn,Xτn) = ϕ(s, x) +

∫ τn

s

Aurϕ(r ,Xr) dr +

∫ τn

s

ϕx(r ,Xr)σur (r ,Xr) dBr

74

+

∫ τn

s

∫R[ϕ(r ,Xr− + γ(r ,Xr−, z))− ϕ(r ,Xr−)] N(dr , dz) (4–5)

We would now like to show that the two martingale integrals go to zero in expectation,

the term with the integral with respect to Brownian motion can be treated in the same

manner as before. Hence our focus will be on the term involving the integral with respect

to the martingale N, we would like to show this term goes to zero as well. To do this we

must show that the following integrability condition is satisfied

E s,x[∫ τn

s

∫R[ϕ(r ,Xr− + γ(r ,Xr−, z))− ϕ(r ,Xr−)]

2 ν(dz) dr

]< ∞

To do this we must use the fact that ϕ is C 1 on [s, τn] hence is Lipschitz continuous so

that there exists a constant K > 0 such that

|ϕ(r ,Xr− + γ(r ,Xr−, z))− ϕ(r ,Xr−)| ≤ K |γ(r ,Xr−, z)|

Using this fact and the additional hypothesis we may now show that the integrability

condition is satisfied

E s,x[∫ τn

s

∫R[ϕ(r ,Xr− + γ(r ,Xr−, z))− ϕ(r ,Xr−)]

2 ν(dz) dr

]≤ K 2E s,x

[∫ τn

s

∫R|γ(r ,Xr−, z)|2 ν(dz) dr

]≤ K 2E s,x

[Cγ

∫ τn

s

(1 + |Xr−|2 + ||u||2) dr]< ∞

With this integrability condition satisfied we have by a standard result of martingales that

E s,x[∫ τn

s


]= 0

Hence taking expectations in Equation (4–5) we get that

E s,x [(τn,Xτn)] = Es,x

[ϕ(s, x) +

∫ τn

s

Aurϕ(r ,Xr) dr

]From this point on the proof is the same as the proof of Theorem 2.3, the reader should

consult this theorem for further details.

75

Armed with a verification theorem in the case for Levy processes with jumps we may

now proceed with the same algorithm as before to solve the optimal control problem.

The basic methodology of the algorithm is the same as the case without jumps, however

the introduction of a discontinuous part of the Levy process will make the analysis more

interesting. The additional complication of a jump process will not hinder a closed form

solution in the power utility case, hence we will be able to directly compare the two

cases. The comparison between the two cases will be the main focus for the remainder

of the paper, we develop the remaining structure needed for a solution then proceed with

the main results.

76

CHAPTER 5OPTIMAL CONTROL PROBLEM IN THE JUMP CASE

5.1 Classic Merton Problem with Jumps

In this section we extend the classical solution of the optimal control problem from

the Merton case to include a price process with jumps. We will inherit much of the

same notation as the no jump case, as much of the work to follow is a generalization of

that case to include processes with jumps. We will consider both the finite and infinite

horizon cases and show that the finite horizon case is just a generalization of the infinite

horizon case. However, we first begin our analysis naively assuming we do not know

this for certain, hence we will consider the infinite horizon case and then work our way

to more general cases. Let (Ω,F ,F[0,∞),P) be a filtered probability space. We will again

assume that we may apply the mutual fund theorem so that our market consists of a risk

free asset and a risky asset. The risk free Rt asset evolves according to the differential

equation

dRt = rRt dt; R0 = 1

where r ≥ 0 is a constant which represents the risk free rate of interest. The risky asset

St evolves according to a Geometric Levy process which is essentially a Geometric

Brownian motion model with an added integral for the jump/discontinuous part. The

differential form of the this process is given by

dSt = αSt dt + σSt dBt + St−

∫ ∞

−1zN(dt, dz); S0 = s0

where α ≥ 0 and σ ≥ 0 represent the rate of the return and volatility of the asset St

respectively. With this assumed form of the Geometric Levy process for the risky asset

we have that γ(s, z) = z , to have a unique solution to this stochastic differential equation

we must have that 1 + γ(s, z) > 0 which implies that z > −1. This means we may only

assume jump sizes that are larger than −1, so we may have jumps in both directions but

negative jumps cannot be too large. To include larger negative jumps one may change

77

the model of the risky asset, we do not address that in this paper and will restrict to

jumps larger than −1. Assuming we have a self financing trading strategy as before we

have that the wealth of the portfolio evolves according to

dWt = (α− r)θtWtdt + [rWt − ct ]dt + σθtWtdBt + θtWt−

∫ ∞

−1zN(dt, dz) (5–1)

with initial conditionW0− = w . In order for Equation (5–1) to satisfy the proper

measurability conditions we note that we are usingWt− rather thanWt , however since

Wt is a cadlag process we have thatWt is the same asWt−. Once we have written

down the wealth process explicitly which serves as the constraint in our optimization

problem, we proceed to restate the cost functional in the power case that will be used for

the optimization.

Jθ,c(s,w) = E s,w[∫ ∞

s

e−δt cγt

γdt

]= e−δsE y

[∫ ∞

0

e−δt cγs+t

γdt

]Given this wealth process and cost functional we would like to find optimal controls c∗,

θ∗, and Φ(s,w) such that

Φ(s,w) = supθ,cJθ,c(s,w) = Jθ

∗,c∗

To solve the optimization problem we proceed in the same fashion as the no jump case,

in particular we first write down the Hamilton Jacobi Bellman equation using Theorem

4.9. The Hamilton Jacobi Bellman equation in the jump case is given by Equation (5–2),

this is essentially the same as the case without jumps with the addition of an integral

term. The integral term is independent of the consumption process c , however this term

contains an explicit dependence on θ. The explicit dependence on theta adds a layer of

complexity not seen in the case without jumps, it is this added complication that makes

the jumps case interesting for our research. We proceed in finding the optimal controls

c∗ and θ∗ given the Hamilton Jacobi Bellman, once found we will write down the explicit

value function ϕ(s,w) and use the verification theorem to show that the optimal control

78

problem is solved.

supθ,c

[e−δs c

γ


1

2σ2w 2θ2ϕww (5–2)

+

∫ ∞

−1[ϕ(s,w(1 + θz))− ϕ(s,w)− ϕw(s,w)θwz ] ν(dz)

]= 0

The optimal solution for the consumption rate is the same as the case without jumps and

is given by

c∗ =(e−δsϕw

) 1γ−1

However the the optimal control for θ presents much more of a challenge in this case, in

particular we are not able to find θ∗ explicitly unless we have more information about the

given Levy process. The solution for θ∗ in the jump case is a solution of the non linear

Equation (5–3), where the nonlinearity comes from the extra integral term not present in

the case without jumps.

(α− r)wϕw + σ2w 2θϕww

+∂

∂θ

∫ ∞

−1[ϕ(s,w(1 + θz))− ϕ(s,w)− ϕw(s,w)θwz ] ν(dz) = 0 (5–3)

Since this term involves an integral with respect to Levy measure the explicit solution

requires knowing the specific Levy measure ν(dz) to find an explicit solution for θ∗.

However, we may make considerable progress in our analysis before choosing a Levy

measure if we assume the value function has the same form as before. In particular,

assuming the value function has the form ϕ(s,w) = K jp0 e−δsw γ we may write the

equation involving θ∗ as

(α− r)wK jp0 e−δsγw γ−1 + σ2w 2θK jp0 e−δsγ(γ − 1)w γ−2 (5–4)

+∂

∂θ

∫ ∞

−1

[K jp0 e

−δsw γ(1 + θz)γ − K jp0 e−δsw γ − θwzK jp0 e−δsγw γ−1

]ν(dz) = 0

79

We may factor out the common terms K jp0 e−δsw γ from this equation to get

K jp0 e−δs

[(α− r)γ + σ2θγ(γ − 1) + ∂

∂θ

∫ ∞

−1[(1 + θz)γ − 1− θzγ] ν(dz)

]w γ = 0

Since we are assuming that w > 0 and K jp0 = 0 we may divide out the factor K jp0 e−δsw γ

to get a simplified form of this equation. Moving the derivative inside of the integral the

condition for θ∗ becomes

(α− r)γ + σ2θγ(γ − 1) +∫ ∞

−1

[γ(1 + θz)γ−1z − γz

]ν(dz) = 0

since γ ∈ (0, 1) we may factor out a γ and divide it out to get

(α− r) + σ2θ(γ − 1) +∫ ∞

−1

[(1 + θz)γ−1 − 1

]zν(dz) = 0 (5–5)

Although an explicit solution to Equation (5–5) is not possible until we specify a Levy

measure we may derive a condition under which a solution of this equation exists by

using the Mean value theorem. In fact although finding a solution to Equation (5–5) is

possible once given a Levy measure, it turns out that many of these equations will be

a nonlinear in θ which leads to closed form solutions in only a small number of cases.

Although some closed form solutions for θ∗ were found we will not provide those results

here, instead we use numerical algorithms to find the solutions to this equation for many

of the cases listed below. To proceed with finding a condition that allows a solution to

Equation (5–5) we define the function

η(θ) = (α− r)− σ2θ(1− γ)−∫ ∞

−1

[1− (1 + θz)γ−1

]zν(dz)

Then we have that η(0) = α − r > 0, using the fact that η(θ) is continuous in θ a solution

θ∗ ∈ (0, 1] exists if η(1) ≤ 0 which happens exactly when

α− r ≤ σ2(1− γ) +

∫ ∞

−1

[1− (1 + z)γ−1

]zν(dz) (5–6)

80

So given a Levy measure we may use this condition to check for the existence of an

optimal control θ∗ in the jump case, note if ν = 0 we do recover θ∗ in the classical case

as should be the case. We also see that Equation (5–5) does not depend on depend on

the wealth w so that there is no explicit time dependence just as in the classical case.

Assuming the optimal solution for θ∗ = θ1 we may explicitly find the constant K jp0 , much

of this work has already been done we need only make minor changes to the results

above. Putting the optimal controls c∗ = (K jp0 γ)1

γ−1w and θ∗ = θ1 into the Hamilton Jacobi

Bellman Equation (5–25) we get

1

γe−δs(K jp0 γ)

γγ−1w γ − δK jp0 e

−δsw γ + θ1(α− r)wK jp0 e−δsγw γ−1 + rwK jp0 e−δsγw γ−1

−(K jp0 γ)1

γ−1wK jp0 e−δsγw γ−1 +

1

2σ2w 2θ21K

jp0 e

−δsγ(γ − 1)w γ−2

+K jp0 e−δs

[(α− r)γ +

∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

]w γ = 0 (5–7)

Combining like terms and factoring out the common term K jp0 e−δsw γ we have

K jp0 e−δs

[(K jp0 γ)

1γ−1 (1− γ)− δ + θ1(α− r)γ + rγ + 1

2σ2θ21γ(γ − 1)

+

∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

]w γ = 0

which gives that

(K jp0 γ)1

γ−1 (1− γ) = δ − αθ1γ − r(1− θ1)γ − 12σ2θ21γ(γ − 1)

−∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

solving for K jp0 we get

K jp0 =1

γ

[1

1− γ

(δ − αθ1γ − r(1− θ1)γ − 1

2σ2θ21γ(γ − 1)

−∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

)]γ−1

81

hence we have the value function in this case is given by

ϕ(s,w) =1

γ

[1

1− γ

(δ − γr − (α− r)θ1γ − 1

2σ2θ21γ(γ − 1)

−∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

)]γ−1e−δsw γ (5–8)

As in the non jump cases we have that the constant is being raised to the γ − 1 power

so the solution is only valid if the term in brackets of Equation (5–8) is positive. For this

term to be positive we must have that

δ > γr + (α− r)θ1γ +1

2σ2θ21γ(γ − 1) +

∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz) (5–9)

Before writing the value function in a more compact notation we perform a quick check

of consistency of Equation (5–8) by letting ν = 0 so that the integral term is removed and

θ1 =α−r

σ2(1−γ)so we may combine the terms involving θ1 to get

− (α− r)θ1γ − 12σ2θ21γ(γ − 1) = − (α− r)2γ

2σ2(1− γ)

so that the value function in the jump case Equation (5–8) is the same as the no jump

case Equation (3–11). To simplify the notation we may take all the terms in parentheses

that involve θ1 and the term γr which are all constant in s,w and set

χ = γr + (α− r)θ1γ +1

2σ2θ21γ(γ − 1) +

∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz) (5–10)

hence the value function becomes

ϕ(s,w) =1

γ

[δ − χ

γ − 1

]γ−1e−δsw γ (5–11)

Given this simplified form of the value function we may take the necessary derivative ϕw

and plug into the equation for c∗ to find that

c∗ =δ − χ

γ − 1w

82

so that the optimal controls for this version of the problem are

ct =δ − χ

γ − 1Wt θt = θ1 (5–12)

Plugging these optimal controls into Equation (3–5) we find that the wealth process is a

Geometric Levy process of the form

dWtWt=

[(α− r)θ1 + r −

δ − χ

(1− γ)

]dt + σθ1 dBt + θ1

∫ ∞

−1zN(dt, dz) W0 = w

This equation has a unique strong solution just as in the non jump case, to show that

the value function in Equation (5–8) is to optimal solution we need to show the other

conditions of the verification theorem are satisfied. We first begin by showing that the

control

ut =

(δ − χ

γ − 1Wt , θ1

)is admissible i.e. ut ∈ A[s,∞). The control ut is measurable and Ft adapted, the check

for integrability of the control process is similar to previous cases

E s,w[∫ ∞

s

||ut ||2 dt]= E s,w

[∫ ∞

s

(δ − χ

γ − 1Wt

)2+ θ21

dt

]

=

(δ − χ

γ − 1

)2E s,w

[∫ ∞

s

W 2t dt

]+ θ21Ps,w [s,∞) < ∞

The integrability condition for the cost functional is similar to the non jump case so the

complete details are left out, the summary is as follows

Jc,θ(s,w) ≤ 1γ

(δ − χ

γ − 1

)γ (Ps,w [s,∞) + E s,w

[∫ ∞

s

W 2t dt

])< ∞ (5–13)

Hence all the conditions of admissibility have been met in this case so the control

ut ∈ A[s,∞), we now proceed to check the remaining conditions of the verification

theorem. We have that the function ϕ(s,w) ∈ C 1,2 is continuous, and satisfies the

Hamilton Jacobi Bellman equation by construction so we only need to check the growth

83

condition which can be done using the same argument as before

|ϕ(s,w)| =

∣∣∣∣∣1γ[δ − χ

γ − 1

]γ−1eδsw γ

∣∣∣∣∣ ≤ 1γ[δ − χ

γ − 1

]γ−1|w |γ ≤ 1

γ

[δ − χ

γ − 1

]γ−1(1 + |w |2)

The remaining conditions on f c,θ(s,w) and σc,θ(s,w) have already been checked or can

be checked with similar arguments, hence the function ϕ(s,w) satisfies the verification

theorem so that ϕ(s,w) = Φ(s,w). This means the function ϕ(s,w) in Equation (5–11)

is the solution to the optimization problem, the solution presented is independent of the

Levy measure. We provide numerical analysis of this solution in the numerical results

section of the paper, we will try to write down the explicit value functions and provide the

corresponding plots.

5.2 Jump Case with Log Utility

We now consider the case where the utility function is of logarithmic form, it turns

out that this case is just an extreme case of a power utility. However, the log model

presents some roadblocks that hinder a complete analysis as performed in the power

utility case. In particular, I have not been able to solve the finite horizon case for

logarithmic utility. It also turns out that the numerical results are not as robust as the

power utility case, hence we will not perform as extensive of an analysis. This section

is mostly presented for completeness of the work that I have done up to this point and

this section may serve as a precursor to future work along these lines. We begin the

analysis with the cost functional and proceed along the same lines as before, hence the

explanations will be sparse.

Jθ,c(s,w) = E s,w[∫ ∞

s

e−δt log ct dt

]= e−δsE y

[∫ ∞

0

e−δt log cs+t dt

](5–14)

We will consider the same portfolio as in the power function case above, in particular

the risky asset is driving by the same Geometric Levy process so that the wealth of the

portfolio evolves according to Equation (5–1). Given the new form of the utility function

the Hamilton Jacobi Bellman changes only slightly (the only difference is in the first

84

term) and now has the following form

supθ,c


1

2σ2w 2θ2ϕww (5–15)

+

∫ ∞


]= 0

To simplify the analysis later we would like to remove the e−δs being multiplied by the

log c term, to do this we define an auxiliary function ϕ and solve it’s Hamilton Jacobi

Bellman equation. Since we are working over an infinite horizon we may again write

ϕ(s,w) = e−δs ϕ(0,w), this gives us a new form of the Hamilton Jacobi Bellman equation

in terms of ϕ given by

supθ,c

[log c − δϕ+ θ(α− r)w ϕw + (rw − c)ϕw +

1

2σ2w 2θ2ϕww (5–16)

+

∫ ∞

−1

[ϕ(w(1 + θz))− ϕ(w)− ϕw(w)θwz

]ν(dz)

]= 0

so that the first order condition for the consumption leads to

1

c− ϕw = 0 =⇒ c∗ =

1

ϕw(5–17)

The second difference is found when looking at the optimal fraction of wealth in the

risky asset. The optimal θ∗ still satisfies Equation (5–3) and as before to perform

further analysis on the existence of an optimal control we must assume the form

of the value function. In this case the value function is assumed to have the form

ϕ(w) = (K jl0 logw + Kjl1 ). Given this functional form of the value function we may take a

closer look at Equation (5–3), plugging in ϕ we get

(α− r)wK jl01

w+ σ2w 2θK jl0

−1w 2+

∂

∂θ

∫ ∞

−1

[K jl0 logw(1 + θz)− K jl0 logw − θwzK jl0

1

w

]ν(dz) = 0

Factoring out K jl0 and simplify the expression we find that

K jl0

[(α− r)− σ2θ +

∂

∂θ

∫ ∞

−1[log (1 + θz)− θz ] ν(dz)

]= 0

85

Bringing the partial derivative into the integral we get

η1(θ) =: (α− r)− σ2θ −∫ ∞

−1

[1− 1

1 + θz

]zν(dz) = 0 (5–18)

As before we have that η1(0) = α − r > 0 so if η1(1) ≤ 0 using the continuity of η1 and

the Mean value theorem we have that a θ∗ ∈ (0, 1] exists, i.e

(α− r) ≤ σ2 +

∫ ∞

−1

[1− 1

1 + z

]zν(dz) (5–19)

Let the optimal control for the portfolio weight be given by θ∗ = θ2, then we are able to

find the constants K jl0 and K jl1 explicitly. Using the fact that c∗ = w

K jl0and θ∗ = θ2 we may

plug these back into the Hamilton Jacobi Bellman Equation (5–16)

log

(w

K jl0

)− δ(K jl0 logw + K

jl1 ) + θ2(α− r)wK jl0

1

w+ rwK jl0

1

w− wK jl0K jl01

w

+1

2σ2w 2θ22K

jl0

−1w 2+

∫ ∞

−1

[K jl0 log(w(1 + θ2z))− K jl0 logw − θ2wzK

jl0

1

w

]ν(dz) = 0

Simplifying the expression we get that

log

(w

K jl0

)− δK jl0 logw − δK jl1 + θ2(α− r)K jl0 + rK

jl0 − 1−

1

2σ2θ22K

jl0

+K jl0

∫ ∞

−1[log(1 + θ2z)− θ2z ] ν(dz) = 0

which we may rewrite as

(1− δK jl0 ) logw +

(− logK jl0 − δK jl1 + θ2(α− r)K jl0 + rK

jl0 − 1−

1

2σ2θ22K

jl0 (5–20)

+K jl0

∫ ∞

−1[log(1 + θ2z)− θ2z ] ν(dz)

)= 0

The term in parentheses in Equation (5–20) is constant in w , since logw is independent

of the constant term we must have that the scalar multiple of the logarithm term must be

zero. This gives us that K jl0 =1δ, since the constant term must also be zero to get a zero

sum we can also solve for the constant K jl1 . Isolating the constant K jl1 and factoring out a

86

1δ

we find

K jl1 =1

δ2

[δ log δ + θ2(α− r) + r − 1− 1

2σ2θ22 +

∫ ∞

−1[log(1 + θ2z)− θ2z ] ν(dz)

](5–21)

Hence the value function is just an extension of the case with no jumps and has the form

ϕ(s,w) = e−δs

(1

δlogw +

1

δ2

[δ log δ + θ2(α− r) + r − 1− 1

2σ2θ22 (5–22)

+

∫ ∞

−1[log(1 + θ2z)− θ2z ] ν(dz)

])The optimal consumption process is the same as the case with no jumps c∗ = δw while

the optimal choice for the fraction of wealth of the risky asset is θ∗ = θ2. Hence for

general times we have that the optimal controls are

ct = δWt θt = θ2

hence the wealth equation is a Geometric Brownian motion of the form

dWtWt= [(α− r)θ2 + r − δ] dt + σθ2 dBt + θ2

∫ ∞

−1zN(dt, dz) W0 = w (5–23)

As a check for consistency between the jump and no jump case we note that if ν = 0

and θ2 =(α−r)σ2

then we have that

θ2(α− r)− 12σ2θ22 =

(α− r)2

2σ2

so that the value function for the jump case reduces to the no jump case Equation

(3–21). This will be the extent of our analysis in the log utility infinite horizon case, the

reason for this is that using same parameters as before the conditions for the existence

of an optimal θ∗ are not satisfied. It turns out that for the condition η(1) ≤ 0 to be

satisfied we must use values of α, r ,σ that are not feasible in practice. In particular if

we consider the Poisson case with a jump of size one at z = 1 then the equation for η

Equation (5–18) gives that (α − r) − σ2 + 12≤ 0. Since (α − r) > 0 the only way for this

equation to hold is if σ2 − 12> 0 but this only holds when σ > 1/

√2 i.e. when σ /∈ [0, 1].

87

If one were considering a more general problem outside of the financial setting we

could use values of sigma outside of this range and present the results, however we

are not currently concerned with this more general setting so we omit any results in this

direction.

5.3 Finite Horizon Power Utility with Jumps

Having completed the analysis on the log utility case we move back to the case

of power utility, and perform the analysis that is the central results of the paper. In

particular we derive a closed form solution to the optimization problem over a finite

horizon and show that we may recover the solution to the infinite horizon case as

T → ∞. Once we have found the closed form solution in this case we will then move

onto to a more general cost functional presented in this section and derived a closed

form solution in that case as well. After generalizing the results in this section we will

show that the new results are consistent with what we find in this section. The case of

power utility with jumps over a finite horizon, will follow many of the same procedures as

before so we again will be sparse with the explanations of the general procedure. The

cost functional is of the following form


e−δt cγt

γdt

]= E y

[∫ T−s

0

e−δ(s+t) cγs+t

γdt

](5–24)

With this cost functional the Hamilton Jacobi Bellman equation becomes

supθ,c

[e−δs c

γ


1

2σ2w 2θ2ϕww (5–25)

+

∫ ∞


]= 0

with boundary condition ϕ(T ,w) = 0. We note that the Hamilton Jacobi Bellman

equation is the same as the infinite horizon case, however the major difference comes

in the fact that the PDE problem now consists of a boundary condition at the end of the

time horizon. This boundary condition will be used to solve for the constant to arrive at

a particular solution among the infinitely many solutions of the Hamilton Jacobi Bellman

88

equation, before we arrive at the PDE we must find the optimal controls to remove the

supremum. The optimal controls are the same as before, c∗ = (eδsϕw)1

γ−1 and θ∗ is a

solution of the equation

(α− r)wϕw + σ2w 2θϕww +∂

∂θ

∫ ∞

−1[ϕ(s,w(1 + θz))− ϕ(s,w)− ϕw(s,w)θwz ] ν(dz) = 0

Since the cost functional is over a finite horizon we assume the value function is of

the form ϕ(s,w) = h(s)w γ, in particular we are not able to factor out the dependence

completely by removing e−δs from the cost functional. This will make the PDE we are

trying to solve more complicated than the infinite horizon case, but an explicit solution is

still possible. To proceed in that direction we use the fact that the optimal control for θ∗

must satisfy the equation

γh(s)

[(α− r) + σ2θ(γ − 1) +

∫ ∞

−1

[(1 + θz)γ−1 − 1

]zν(dz)

]w γ = 0

hence an optimal control θ∗ = θ1 exists if

α− r ≤ σ2(1− γ) +

∫ ∞

−1

[1− (1 + z)γ−1

]zν(dz) (5–26)

Given this condition is satisfied so that θ∗ exists and that c∗ = (eδsϕw)1

γ−1 we may

write the Hamilton Jacobi Bellman equation as the following integro-partial differential

equation

1− γ

γ(eδs)

1γ−1ϕ

γγ−1w + ϕs + θ1(α− r)wϕw + rwϕw +

1

2σ2w 2θ21ϕww

+

∫ ∞

−1[ϕ(s,w(1 + θ1z))− ϕ(s,w)− ϕw(s,w)θ1wz ] ν(dz) = 0 (5–27)

Given the form of the value function we may find the function h(s) explicitly as before

by taking the necessary derivatives of ϕ on plugging into Equation (5–27) to get the

following ordinary differential equation for h(s),

1− γ

γ(eδs)

1γ−1 (γh(s))

γγ−1w γ +

1

2σ2w 2θ21γ(γ − 1)w γ−2h(s) + θ1(α− r)wγh(s)w γ−1

89

+h′(s)w γ + rwγh(s)w γ−1 +

∫ ∞

−1

[h(s)w γ(1 + θ1z)

γ − h(s)w γ − γθ1wzh(s)wγ−1] ν(dz) = 0

We may simplify this equation by factoring out and dividing by w γ to end up with

h′(s) +

[θ1(α− r)γ + rγ + 1

2σ2θ21γ(γ − 1) +

∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

]h(s)

=γ − 1γ(eδs)

1γ−1 (γh(s))

γγ−1

It turns out that this equation has a particularly nice form once we look past all the

complications involved with the θ∗ term, and it ends up being an Bernoulli ordinary

differential equation in h(s) of the form

h′(s) + p(s)h(s) = q(s)h(s)γ

γ−1 (5–28)

where we have that

χ =

[θ1(α− r)γ + rγ + 1

2σ2θ21γ(γ − 1) +

∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

]=: p(s)

and

q(s) = (γ − 1)(γeδs)1

γ−1

We note that the term χ is independent of the time parameter and is in fact constant,

which makes solving the Bernoulli differential equation slightly easier. To solve Equation

(5–28) we use the general form of the solution for Bernoulli type equations. Using

Equation (3–30) we have that the solution

h(s) =

[−(γ)

1γ−1

κe

δγ−1 s + Ce

χγ−1 s

]1−γ

where κ = δ−χγ−1 . Using the boundary condition h(T ) = 0 we find that the constant is

C = (γ)1

γ−1

κe

δ−χγ−1T which leads to the general solution

h(s) =

[−(γ)

1γ−1

κe

δγ−1 s +

(γ)1

γ−1

κe

δγ−1Te−

χ(T−s)γ−1

]1−γ

90

=κγ−1

γe−δs

[−1 + e

δ−χγ−1 (T−s)

]1−γ

=1

γ

[δ − χ

1− γ

]γ−1e−δs

[1− e


]1−γ

plugging the function h(s) into the equation for ϕ(s,w) we find that the value function in

this case is given by

ϕ(s,w) =1

γ

[δ − χ

1− γ

]γ−1e−δs

[1− e


]1−γ

w γ

We may rewrite the value function by combining the terms in parentheses to get

ϕ(s,w) =1

γ

δ − χ

(1− γ)(1− e


)γ−1

e−δsw γ (5–29)

This solution is only valid if the condition δ > χ is satisfied, in particular we must have

that

δ > θ1(α− r)γ + rγ + 12σ2θ21γ(γ − 1) +

∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

An interesting result to note here is the we end up factoring out e−δs anyway as in the

infinite horizon case, but this does not remove all time dependence. There is still a time

dependence in the term(1− e


)which acts like a discount factor for this case

when compared with the infinite horizon case. To check consistency with the infinite

horizon case we note that if we let T → ∞ in Equation (5–29) we should recover the

infinite horizon case with jumps in Equation (5–11). Just as in the non jump case we

see that this is in fact what happens, we may also provide a numerical justification of

this result by taking larger values of T and presenting the plots. This is done in Figure

5− 1, we see that as the values of T get larger the maximum/minimum values of ϕ(s,w)

approach the values of the infinite horizon case. The plots provided in Figure 5− 1

are for the fixed Levy measure ν(dz) = δ(1), as this is the simplest Levy measure we

consider. One can easily perform the same analysis using a different Levy measure as

91

long as the required integrals can be found. Given that ϕ(s,w) exists and is written in

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

0

1

2

3

4

T= 1max= 4.208099

min= 0

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

T= 2.5max= 4.532554min= 0.5144292

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

T= 5max= 4.550642min= 0.5290862

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

T= 7.5max= 4.550786min= 0.5292013

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

T= 100max= 4.550787min= 0.5292022

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4


Figure 5-1. Plots of the value function for values of T = 1, 2.5, 5, 7.5, 100 compared tothe plot of the value function in the infinite horizon case.

the simplified form of Equation (5–29) we find that the optimal controls are given by

ct =

δ − χ

(1− γ)(1− e

δ−χγ−1T

)Wt θt = θ3

hence the wealth Equation (3–5) is a Geometric Levy process of the form

dWtWt=

(α− r)θ1 + r −δ − χ

(1− γ)(1− e

δ−χγ−1T

) dt + σθ1 dBt + θ1

∫ ∞

−1zN(dt, dz) W0 = w

We may simplify the remaining analysis that involves showing that the control process

ut is admissible for this case, specifically we have already performed an analysis which

only differs slightly with this case. In Section 3.4 we have that the optimal consumption

process ct only differs from this case by the constant being subtracted in the numerator,

92

while the fraction of wealth in both cases are constants. This means that the analysis

used to show that the technical conditions of the verification theorem in Section 3.4 can

be carried over to show that all conditions are satisfied in this case.

5.4 Generalization of Bequest Function in the Jump Case

We may solve the problem from the previous section in a more general setting, by

extending the bequest function to be non zero. The problem we just solved over the

finite horizon was for the case where the bequest function was zero in our assumed cost

functional. A non zero bequest function only changes the problem in the sense that the

boundary condition for the function h(s) changes so that the constant C is more general.

Let λ > 0 be a constant, we will now consider a cost functional of the form


e−δt cγt

γdt + λW γ

T

]= E y

[∫ T−s

0

e−δ(s+t) cγs+t

γdt + λW γ

T−s

]Given this cost function the optimal control problem remains the same as the previous

section in that the value function has the form ϕ(s,w) = h(s)w γ where h satisfies the

Bernoulli equation

h′(s) + p(s)h(s) = q(s)h(s)γ

γ−1 (5–30)

where we have that

χ =

[θ1(α− r)γ + rγ + 1

2σ2θ21γ(γ − 1) +

∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz)

]=: p(s)

and

q(s) = (γ − 1)(γeδs)1

γ−1

and whose general solution is given by

h(s) =

[−(γ)

1γ−1

κe

δsγ−1 + Ce

χsγ−1

]1−γ

The main difference between the general case is that the boundary condition now says

that ϕ(T ,w) = λw γ, which implies that h(T ) = λ. Hence we may begin our new

93

analysis of this problem form this point, the main impact of this change will be the new

value of the constant C . We will find the new constant in this case and then show that

the general case reduces to case from the previous section when λ = 0. Applying the

boundary condition we have that

λ =

[−(γ)

1γ−1

κe

δγ−1T + Ce

χγ−1T

]1−γ

Solving for the constant C by performing the required algebra leads to

C = e−χTγ−1

[(λ)

11−γ +

(γ)1

γ−1

κe

δTγ−1

](5–31)

Plugging this into the equation for h(s) we may find the explicit solution with this more

generalized boundary condition

h(s) =

[−(γ)

1γ−1

κe

δsγ−1 + e−

χTγ−1

[(λ)

11−γ +

(γ)1

γ−1

κe

δTγ−1

]e

χsγ−1

]1−γ

=

[(γ)

1γ−1

κ

(−e

δsγ−1 + e−

χ(T−s)γ−1

[κ

(γ)1

γ−1(λ)

11−γ + e

δTγ−1

])]1−γ

=κγ−1

γ

[−e

δsγ−1 + e−

χ(T−s)γ−1

[κ (λγ)

11−γ + e

δTγ−1

]]1−γ

=1

γ

[δ − χ

1− γ

]γ−1 [e

δsγ−1 − e−

χ(T−s)γ−1

[κ (λγ)

11−γ + e

δTγ−1

]]1−γ

=1

γ

[δ − χ

1− γ

]γ−1e−δs

[1− e−

χ(T−s)γ−1 e

−δsγ−1

[κ (λγ)

11−γ + e

δTγ−1

]]1−γ

=1

γ

[δ − χ

1− γ

]γ−1e−δs

[1− e


[κ (λγ)

11−γ e

−δTγ−1 + 1

]]1−γ

If we now combine the terms in brackets to a single power we find that

h(s) =1

γ

δ − χ

(1− γ)(1− e


[δ−χ1−γ (λγ)

11−γ e

−δTγ−1 + 1

])γ−1

e−δs

94

Given that δ > χ so that h(s) is well defined we have that the value function is given by

ϕ(s,w) =1

γ

δ − χ

(1− γ)(1− e



11−γ e

−δTγ−1 + 1

])γ−1

e−δsw γ (5–32)

Equation (5–32) is the most general value function in the power utility case we have

derived, each of the two previous cases can be recovered from Equation (5–32). First,

letting λ = 0 we may recover Equation (5–29) since the term[δ−χ1−γ (λγ)

11−γ e

−δTγ−1 + 1

]=

1, a result that provide some reassurance to the correctness of the equations.

Furthermore, letting λ = 0 and T → ∞ in Equation (5–32) we recover Equation

(5–11) since(1− e



11−γ e

−δTγ−1 + 1

])→ 1, again a reassuring result.

From a more numerical standpoint we get consistency in the closed form solution

presented above by comparing the plots of Equation (5–32) with the case where λ = 0.

In particular we begin with positive values of λ and approach zero from above, from a

theoretical standpoint we should have that plots with positive lambda should become

more like the plot with λ = 0 when we are close enough to zero. The results of the plots

of this process is presented in Figure 5− 2 (again this plot is for a Levy measure of the

form ν(dz) = δ(1) for simplicity) and we see that the numerical results are consistent

with what we would expect theoretically. In particular, one λ = 1e − 9 the values of the

maximum/minimum are the same as the case when λ = 0, so that Figure 5− 2 models

the behavior of the infinite horizon case correctly. This provides yet another check of

consistency to the closed form solution we have derived in this section and puts the

validity of the is formula on solid ground. We note that this equation is a product of the

assumption of geometric Levy market model, one may consider more complex models

for the risky asset which may lead to different formulas. The assumption of a geometric

Levy model consistently leads to a formula of this form which hinting that a completely

general formula may be possible, this is left for future research.

95

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

0.5

1.0

1.5

2.0

2.5

lambda= 1max= 2.645509min= 0.0968893

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

lambda= 0.5max= 3.876951min= 0.1024662

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

lambda= 0.25max= 4.127803min= 0.0622882

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

1

2

3

4

lambda= 0.125max= 4.188169

min= 0.03312922

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

0

1

2

3

4

lambda= 1e−09max= 4.208099

min= 0

s

0.00.2

0.40.6

0.8

1.0

w

2

4

68

10

Phi(s,w

)

0

1

2

3

4

lambda=0max= 4.208099

min= 0

Figure 5-2. Plots of the value function for values of λ = 1, .5, .25, .125, 1e-09 comparedto the plot of the value function for λ = 0

96

CHAPTER 6NUMERICAL RESULTS AND CONCLUSION

6.1 Jump diffusion and Levy triplets

The four diffusion models we consider throughout the paper are all examples of the

general Merton jump diffusion model presented in [19]. They provide a range of jump

characteristics that can be found by considering the characteristic functions of the Levy

process associated with the jump part of the the diffusion model. In this section we will

present the different jump characteristics of each of the models and then provide explicit

numerical schemes to compute the values of θ∗ in each case. We first consider the

Poisson process which has exponentially distributed jumps and only has jumps of size

one. To construct the Poisson process let τk∞k=1 be independent exponential random

variables with parameter λ i.e.

P(τk ≥ y) = e−λy

Let Tn =∑nk=1 τk then the process πt =

∑∞n=1 1(t≥Tn) is called the Poisson process with

parameter λ. The jumps of the Poisson process occur at the times Tn and the waiting

times between jumps is exponential distributed. The paths of the Poisson process πt are

cadlag and for each t > 0 has Poisson distribution with intensity λt i.e.

P(πt = n) =(λt)n

n!e−λt n = 0, 1, 2 · · ·

A simple calculation shows that the characteristic function of the Poisson process is

given by

E [e iuπt ] = eλt(eiu−1)

using the notation of Theorem 4.5 the Levy exponent is η(u) = λ(e iu − 1) which has Levy

triplet (0, 0, ν(dz) = δ(1)). The Poisson process case greatly simplifies all numerical

calculations involving ν(dz), however a process with jump size restricted to unity is not

a robust model of stock prices. Hence, we would like to generalize this idea to allow

for processes with varying jump sizes, in particular we want have a distribution of jump

97

sizes. The generalization of the Poisson process is given by the compound Poisson

process defined by

Definition 6.1. Let (Zn)n∈N be a sequence of i.i.d random variables taking values in R

with common distribution νZ1 = νZ and πt a Poisson process of intensity λ independent

of each Zn the the compound Poisson process (Yt)t≥0 is defined by

Yt =

πt∑i=1

Zi

The compound Poisson process retains the exponential waiting times between jumps

however we now have that the jump sizes have the distribution νZ . The characteristic

function of the compound Poisson process has Levy exponent

η(u) = λ

∫R(e iuz − 1)ν(dz)

hence the Levy triplet is (0, 0, ν(dz)). The compound Poisson process now allows for

jumps of arbitrary size choose from a given distribution, we note that when ν(dz) = δ(1)

we do in fact recover the Poisson process. It is also the basis of the Merton jump

diffusion model that we are using to model the risky asset, it’s general form is

Xt = αt + σBt +

∫RzN(t, dz) = αt + σBt + Yt

since∫R zN(t, dz) has a compound Poisson distribution. The characteristic function for

Xt has Levy exponent

η(u) = iαu − 12σ2u2 + λ

∫R(e iuz − 1)ν(dz)

and Levy triplet given by (α,σ2, ν(dz)). If Xt is a compound Poisson process then we

have that there are a finite number of jumps on each time interval which gives that

ν(R) < ∞ and we may show ν(dz) = f (z) dz for some density function of jump

sizes. However, the Levy measure ν does not necessarily need to have a finite number

of jumps on each time interval, as a legitimate Levy measure need only satisfy the

98

weaker condition∫R(z

2 ∧ 1)ν(dz) < ∞. In particular we will consider the measure

ν(dz) = z−1e−z1(z>0) dz which has infinite activity. A Levy process with infinite activity

still provides for a tractable model, since many of the jump sizes are small and only a

finite number have absolute value larger than any given positive number.

6.2 Optimal θ∗ for Power Utility

We would like to provide some numerical results for the cases we solved above.

In particular we would like to compare the value of the optimal portfolio weights θ∗

for different Levy measures. General Levy processes may have a complicated Levy

measure associated which makes find θ∗ difficult if not impossible. However if we restrict

our study to the case where the Levy measure is absolutely continuous with respect to

Lebesgue measure we are able to provide tangible results. Throughout this section we

will restrict our attention to the case of the power utility, so we would like to solve for θ in

the following the equation

(α− r)− σ2θ(1− γ)−∫R[1− (1 + θz)γ−1]zν(dz) = 0 (6–1)

This equation is non linear in θ and the difficulty comes from the integral term,

theoretically we are able to solve this equation for any Levy measure. We recall a

Levy measure is any nonnegative measure ν(dz) on R satisfying ν(0) = 0 and∫R(z2 ∧ 1)ν(dz) < ∞

I will present a few Levy measures which allow us to rewrite Equation (6–1) in terms

of known statistical distributions so that the numerical work is greatly simplified. The

Gamma(a,b) process is a stochastic process (Γt)t≥0 such that for each fixed t the

random variable Γt has a Gamma(a,b) distribution with density

fΓt(z) =bat

Γ(at)zat−1e−bz1(z>0)

99

The Gamma(a,b) process is a pure jump non decreasing (a.s.) Levy process with Levy

measure ν(dz) = az−1e−bz1(z>0) where a, b > 0. This means that jumps whose size

are in the interval [z , z + ∆z ] occur as a Poisson process with intensity ν(dz), where the

parameter a controls the rate of jump arrivals and b controls the jump sizes. Given this

Levy measure we may simplify Equation (6–1) so that we may carry out the numerical

algorithm to find the optimal θ∗.

6.2.1 Γ(1, 1) Process

We consider the case of the Γ(1, 1) Levy process whose Levy measure is given by

ν(dz) = z−1e−z1(z>0)

This is in fact a Levy measure since it satisfies the converse of the Levy-Khintchine

theorem for the existence of a Levy process, we need only show that∫Rmin(1, z2)ν(dz) =

∫ ∞

0

min(1, z2)z−1e−z dz =

∫ 10

ze−z dz +

∫ ∞

1

z−1e−z dz < ∞ (6–2)

The first integral in Equation (6–2) is finite by just a simple integration by parts, the

second integral is a know integral called the exponential integral whose value is given by

Ei(1) = 1.89511781... The finiteness of this quantity guarantees the existence of a Levy

process corresponding to this Levy measure, that Levy process is the Γ(1, 1). With this

Levy measure the integral equation for θ∗ becomes∫R

[1− (1 + θz)γ−1

]zν(dz) =

∫ ∞

0

[1− (1 + θz)γ−1

]e−z dz

=

∫ ∞

0

e−z dz −∫ ∞

0

(1 + θz)γ−1e−z dz

= 1−∫ ∞

0

(1 + θz)γ−1e−z dz

We may now rewrite the remaining integral using the substitution u = (1 + θz) to get∫ ∞

0

(1 + θz)γ−1e−z dz =e1/θ

θ

∫ ∞

1

uγ−1e−u/θ du

100

=e1/θ

θ

[∫ ∞

0

uγ−1e−u/θ du −∫ 10

uγ−1e−u/θ du

]=e1/θ

θ

[θγΓ(γ)−

∫ 10

uγ−1e−u/θ du

]=e1/θ

θ

[θγΓ(γ)− θγ

∫ 1/θ0

tγ−1e−t dt

](t = u/θ)

=e1/θ

θ[θγΓ(γ)− θγΓ(γ)P(1/θ, γ)]

= θγ−1e1/θΓ(γ) [1− P(1/θ, γ)]

where we have used a predefined statistical function from the program R to rewrite the

integral, the function is called the pgamma function which is defined by

P(a, x) = 1/Γ(a)

∫ x1

ta−1e−t dt

Combining these results we have that Equation (6–1) becomes

(α− r)− σ2θ(1− γ)− 1 + θγ−1e1/θΓ(γ)[1− P(1/θ, γ)] = 0

This equation is highly non linear and cannot be solved explicitly, however written in this

form we may easily implement a numerical algorithm using the program R.

6.2.2 Compound Poisson Process with Exponential Density

We will consider the special case when each Zn is exponentially distributed, in particular

we consider the exp(1) distribution whose density function is given by

fZ(z) = e−z1(z>0)

In this case the Levy measure for the compound Poisson process Yt is given by

ν(dz) = λfZ(z) dz = λe−z1(z>0) dz

101

Given the Levy measure we may proceed in a similar fashion as before to calculate the

integral term in Equation (6–1)∫R[1− (1 + θz)] z ν(dz) = λ

[∫ ∞

0

ze−z dz −∫ ∞

0

(1 + θz)γ−1z e−z dz

]= λ

[1−

∫ ∞

0

(1 + θz)γ−1z e−z dz

]Before moving on to the evaluation of the integral we write a general formula for the

type of integral from the previous example since we will encounter it twice more in this

example. Let a, b > 0 be constants, then∫ ∞

1

ua−1e−u/b du =

∫ ∞

0

ua−1e−u/b du −∫ 10

ua−1e−u/b du

= baΓ(a) [1− P(1/b, a)] =: DP(1/b, a) (6–3)

We may again rewrite this integral in a form that makes it easier to evaluate numerically,

to do this we again make the substitution u = 1 + θz∫ ∞

0

(1 + θz)γ−1e−zz dz =e1/θ

θ2

∫ ∞

1

[uγ − uγ−1]e−u/θ du

=e1/θ

θ2

[∫ ∞

1

uγe−u/θ du −∫ ∞

1

uγ−1e−u/θ du

]=e1/θ

θ2[θγ+1Γ(γ + 1)(1− P(1/θ, γ + 1))− θγΓ(γ)(1− P(1/θ, γ))

]= θγ−2e1/θ [θγΓ(γ)DP(1/θ, γ + 1)− Γ(γ)DP(1/θ, γ)]

Using this simplified form of the integral we may write the equation for the optimal

control θ∗ where we will assume λ = 1 for explicit computation

(α− r)− σ2θ(1− γ)− 1 + θγ−2e1/θΓ(γ) [θγDP(1/θ, γ + 1)−DP(1/θ, γ)] = 0

Once we have formulated the each of the the Levy integrals that allows us to run

numerical simulations we compare the values of θ∗ while varying different parameter

values and present the results below. The analysis we perform in this section will be for

the case over the infinite horizon but could be performed in the other cases as well. We

102

use this case so that we are only varying the control θ∗ and not the values of λ or T .

We present the results of this analysis in Tables 6− 1 and 6− 2. Each table contains

the classical no jump case and four cases with jumps where the Levy processes are a

compound Poisson process, a Γ(1, 1), a Poisson process with one jump, and a Poisson

process with two jumps. Table 6− 1 present the values of the optimal θ as we vary the

volatility parameter σ. There are few results from the table which provide a check of

consistency of our analysis, the first is that for a given process an increasing volatility

leads to a decrease θ∗. This result is consistent with what we would expect, as the

volatility increase the risky asset involves more uncertainty hence we should reduce

the fraction of wealth invested in the risky asset and increase the fraction in the safe

asset. The second result is that the fraction of wealth in the risky asset in always larger

in the no jump case when compared to any of the values in the jump cases. This is

consistent with our intuition as the jumps lead to greater uncertainty in the dynamics

of the risky asset, this uncertainty leads to the investor choosing a smaller fraction of

the risky asset in his construction of an optimal portfolio. Finally we note that for fixed

σ the values of θ∗ decrease across each row of the table, this is a little less intuitive

and deserves some further explanation. We begin with the two columns that intuitively

make sense, going from the Poisson case with one jump to the case with two jumps

we expect the value of θ∗ to decrease. This intuitively makes sense as we are adding

an extra jump of the same size to the Levy process which leads to greater uncertainty.

Furthermore, we would expect that going from the Γ(1, 1) process to the Poisson cases

that since the Γ(1, 1) case has a larger number of expected jumps per interval than

the Poisson cases the value of θ∗ should be larger in that case. This intuition is verified

from the table which shows that our model views the Γ(1, 1) case as more risky than

the Poisson cases due to the lower fraction of wealth in the risky asset for the Γ(1, 1)

case when compared to the Poisson cases. The reasoning behind this behavior is that

the Γ(1, 1) contains a larger number of expected jumps on each interval hence this

103

leads to a greater deal of uncertainty in the trajectory of the price process. The Poisson

process contains only one jump of known size so the variation in the trajectory of the

price process is much smaller, hence the investor is willing to put a larger fraction of

the risky asset in the Poisson process when compared to the Γ(1, 1) process. Finally,

we note that the compound Poisson process of density ν(dz) = e−z1(z>0) is the least

”risky” of the Levy processes with jumps in our comparison. An explanation of this

behavior could be that this Levy process always has jumps size less than one on z > 0

since e−z < 1 on this interval. This means that the Levy process associated with this

Levy measure is contains many jumps of small size on each time interval, this behavior

more closely models a continuous price process than any of the other Levy processes

under consideration. Since this is closet to a continuous Levy process we would expect

that the fraction of wealth in the risky asset would be closest to the no jump case, this

expectation is verified in each of the tables below. As another check of consistency of

Table 6-1. Table for θ∗ as a function of σ, where α = .16, r = .06, γ = .5

σ No Jumps e−z1(z>0) Poisson 1 Poisson 2 Γ(1, 1)

0.5 0.80000 0.19062 0.17626 0.14379 0.1057460.51 0.76894 0.18851 0.17456 0.14267 0.1050900.52 0.73964 0.18642 0.17286 0.14156 0.1044300.53 0.71200 0.18434 0.17116 0.14044 0.1037670.54 0.68587 0.18227 0.16947 0.13931 0.1031020.55 0.66116 0.18022 0.16778 0.13819 0.1024330.56 0.63776 0.17818 0.16610 0.13707 0.1017620.57 0.61557 0.17615 0.16442 0.13594 0.1010900.58 0.59453 0.17414 0.16275 0.13482 0.1004150.59 0.57455 0.17215 0.16109 0.13369 0.0997390.6 0.55556 0.17017 0.15944 0.13257 0.099062

Tables 6− 1 and 6− 2 we note that the first row of Table 6− 1 has the same parameters

as the last row of Table 6− 2 hence the values should be the same, a quick glance

at these two rows shows that there is some consistency with the algorithms used to

generate these tables.

6.3 Explicit Value Functions

Once we have found the optimal values of the control θt we write down the explicit

form of the value function for the infinite horizon case using Equation (5–11). To proceed

104

Table 6-2. Table for θ∗ as a function of γ, where α = .16, r = .06, σ = .5

γ No Jumps e−z1(z>0) Poisson 1 Poisson 2 Γ(1, 1)

0.4 0.66667 0.15617 0.14534 0.11883 0.0865450.41 0.67797 0.15905 0.14794 0.12093 0.0881480.42 0.68966 0.16203 0.15063 0.12311 0.0898100.43 0.70175 0.16513 0.15342 0.12536 0.0915360.44 0.71429 0.16835 0.15631 0.12770 0.0933290.45 0.72727 0.17170 0.15932 0.13013 0.0951940.46 0.74074 0.17518 0.16244 0.13265 0.0971340.47 0.75472 0.17880 0.16569 0.13527 0.0991540.48 0.76923 0.18258 0.16907 0.13799 0.1012590.49 0.78431 0.18651 0.17259 0.14083 0.1034540.5 0.80000 0.19062 0.17626 0.14379 0.105746

finding the explicit value functions we will need to compute the constant χ which involves

computing the integral ∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz) (6–4)

where θ1 is the optimal control found in the section above. This integral in the case of

the Poisson point measures is not difficult and has been used throughout the paper, but

we will present them for completeness. The case of a single jump at z = 1 has Levy

measure ν(dz) = δ(1) so the integral in Equation (6–4) becomes∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz) = (1 + θ1)γ − 1− γθ1

which gives rise to an explicit value function. The case of the Poisson process with two

jumps has Levy measure ν(dz) = δ(.5) + δ(1) which gives the integral∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz) = (1 + θ1)γ + (1 + .5θ1)

γ − 2− 1.5γθ1

which again gives rise to a simple form for the value function. The more interesting

cases arise with the other two Levy measures, for example for the Levy measure

ν(dz) = e−z1(z>0) the integral becomes∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz) =

∫ ∞

0

[(1 + θ1z)γ − 1− γθ1z ] e

−zdz

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To evaluate this integral by breaking into three integral, where the second two integral

are easily computed using the fact that∫∞0zne−z dz = n! for all n ∈ N. The first integral

we may compute using the analysis in the previous section for integral involving the

gamma distribution, by making the substitution u = 1 + θ1z .∫ ∞

0

(1 + θ1z)γe−z dz =

e1/θ1

θ1DP(1, γ + 1)

Hence integral in Equation (6–4) becomes∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz) =e1/θ1

θ1DP(1, γ + 1)− 1− γθ1

The final Levy measure we consider is that of the Γ(1, 1) process, we recall this Levy

measure is given by ν(dz) = e−z

z1(z>0), so we need to compute the integral∫ ∞

−1[(1 + θ1z)

γ − 1− γθ1z ] ν(dz) =

∫ ∞

0

z−1 [(1 + θ1z)γ − 1− γθ1z ] e

−z dz

This integral presents a few challenges in terms of finding a simple numerical methodology

at this time, however further work in this direction may provide a workable solution.

In particular there may be available numerical algorithms that allow for efficient

computation, however that is not the main focus of the paper so this is left for future

work. The discrete time Levy measures allow for an adequate analysis of the methodologies

presented in this paper and can be used to check the consistency of the value functions.

6.4 Conclusion

We find that the investment and consumption control problem in the case of

hyperbolic absolute risk aversion power utility U(c) = cγ

γwith stock prices driven by Levy

processes with jumps admits a closed form solution. The closed form solution is found in

both the finite and infinite horizon cases and compared with the Merton solution without

jumps. The formulas for the optimal portfolio weights θ∗ and the the value function ϕ

depend on the Levy process used to model the price of the risky asset in the portfolio,

in particular they depend on the Levy density of the associated Levy measure ν. Given

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four different Levy measures we were able to numerically compute the values of θ∗ and

ϕ so that a comparison could be made to the case without jumps. We also consider

the differences between the four different Levy measures and show that results are

consistent with expectations. The numerical results showed that the optimal weight θ∗

is larger in the case without jumps when compared to any of the Levy jump cases, this

behavior models an investors risk preference when there is greater uncertainty involved

in the underlying price process. The investor is willing to put a larger fraction of the

portfolio in the risky asset when there are no jumps present, and this fraction is reduced

once jumps are factored into the model. As the the size and frequency of the jumps

increase the fraction of wealth in the risky asset decreases, this inverse relationship is

what one would expect in practice. To quantify this result we present tables comparing

the differences in θ∗ between each of the Levy processes, the results show that an

increase in the frequency of jumps leads to decrease in the fraction of wealth an investor

is willing to have in the risky asset. Another interesting result in our analysis was that the

jump diffusion models were less sensitive than the no jump models to changes in the

parameter values. In particular the case without jumps would frequently give values of

θ∗ outside of the region [0, 1] while the jump diffusion model consistently provided values

within this range. A value of θ∗ in [0, 1] is desirable for this particular problem as this

value represent a fraction of total wealth of the portfolio, and the jump diffusion models

seem to capture this feature better than the non jump model. A check of consistency is

provided for each of the closed form solutions of ϕ that we present. The first check of

consistency is the comparison of the formula for ϕ in the finite case without jumps, we

compare the finite horizon case for values of increasing T and show that the graphs of

ϕ for the finite case approach the infinite horizon case as T gets larger. This analysis is

also done in the case with jumps and provide a consistency check of the formula for the

finite horizon case. There is also a check of the final formula presented in which the cost

function depends on the parameter λ. The case for λ = 0 is solved in a previous section

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and this is then compared with different cases for λ > 0 but approaching zero from the

right. The results show that that for small enough λ the values of ϕ become identical

and the models provide the same numerical values. In conclusion, it seems that there

may be some validity to using a jump diffusion model to solve this particular optimal

portfolio choice problem. The presentation of a closed form solution avoids many of the

added technical difficulties that may typically arise from using jump diffusion models.

Hence it seems that the added benefit of using a jump diffusion to better model volatility

skew and the heavy tail of assets returns may be worth the extra effort for the solution

of this particular problem. Future research in this direction could involve using more

complicated Levy processes and hence Levy measure to compute the values of θ∗, this

could entail solving the problem in more than one dimension. One may also be able to

find a general closed form solution for the non linear equation for θ∗ for the power utility

case.

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REFERENCES

[1] D. Applebaum, Levy processes and stochastic calculus, volume 116, CambridgeUniversity Press, Cambridge, second edition, 2009.

[2] S. Asmussen, P. Glynn, Stochastic simulation: Algorithms and analysis, SpringerVerlag, 2007.

[3] R. Bellman, R. Kalaba, Dynamic programming and modern control theory,Academic Press New York, 1965.

[4] J. Bertoin, Levy processes, volume 121, Cambridge University Press, Cambridge,1996.

[5] P. Billingsley, Probability and measure, Wiley-India, 2008.

[6] T. Chan, Pricing contingent claims on stocks driven by Levy processes, Annals ofApplied Probability 9 (1999) 504–528.

[7] G. Chen, G. Chen, S. Hsu, Linear stochastic control systems, CRC, 1995.

[8] R. Cont, P. Tankov, Financial modelling with jump processes, Chapman &Hall/CRC, Boca Raton, FL, 2004.

[9] J. Cox, C. Huang, Optimal consumption and portfolio policies when asset pricesfollow a diffusion process, Journal of economic theory 49 (1989) 33–83.

[10] M.H.A. Davis, A.R. Norman, Portfolio selection with transaction costs, Math. Oper.Res. 15 (1990) 676–713.

[11] W.H. Fleming, T. Pang, An application of stochastic control theory to financialeconomics, SIAM J. Control Optim. 43 (2004) 502–531.

[12] N. Framstad, B. Oksendal, A. Sulem, Optimal consumption and portfolio in ajump diffusion market with proportional transaction costs, Journal of MathematicalEconomics 35 (2001) 233–257.

[13] P. Glasserman, Monte Carlo methods in financial engineering, Springer Verlag,2004.

[14] D. Kannan, V. Lakshmikantham, Handbook of stochastic analysis and applications,CRC, 2002.

[15] I. Karatzas, S. Shreve, Brownian motion and stochastic calculus, Springer, 1991.

[16] F. Longstaff, Optimal portfolio choice and the valuation of illiquid securities, Reviewof financial studies 14 (2001) 407.

[17] H. Markowitz, Portfolio selection, Journal of Finance 7 (1952) 77–91.

109

[18] R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model,J. Econom. Theory 3 (1971) 373–413.

[19] R.C. Merton, Option pricing when underlying stock returns are discontinuous,Journal of financial economics 3 (1976) 125–144.

[20] B. Oksendal, Stochastic differential equations, Springer-Verlag, Berlin, sixth edition,2003. An introduction with applications.

[21] B. Oksendal, A. Sulem, Applied stochastic control of jump diffusions, Springer,Berlin, second edition, 2007.

[22] P.E. Protter, Stochastic integration and differential equations, volume 21,Springer-Verlag, Berlin, second edition, 2004. Stochastic Modelling and AppliedProbability.

[23] L. Rogers, D. Williams, Diffusions, Markov processes, and martingales, Cambridgeuniversity press, 2000.

[24] P. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, TheReview of Economics and Statistics 51 (1969) 239–246.

[25] K.i. Sato, Levy processes and infinitely divisible distributions, volume 68,Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japaneseoriginal, Revised by the author.

[26] W. Schoutens, Levy processes in finance: pricing financial derivatives, Wiley, 2003.

[27] E. Schwartz, C. Tebaldi, Illiquid assets and optimal portfolio choice, NBER Workingpaper (2007).

[28] C. Tapiero, Applied stochastic models and control for finance and insurance, KluwerAcademic Pub, 1998.

[29] J. Yong, X. Zhou, Stochastic controls: Hamiltonian systems and HJB equations,Springer Verlag, 1999.

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BIOGRAPHICAL SKETCH

Ryan Sankarpersad was born in the republic of Trinidad and Tobago in 1981. He

moved to Florida in the early 90’s and has resided there ever since. He graduated the

University of Florida with a Bachelor of Science in mathematics and a Bachelor of Arts

in physics in 2005. He then began graduate work at the University of Florida in the Fall

of 2005 and received his Masters of Science in mathematics in 2007. He completed his

doctorate of philosophy in the area of mathematical finance in the spring of 2011.

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