Model Uncertainty and Robust Duality in Financepinguim.uma.pt/Investigacao/Ccm/icsaa13/page14... ·...

Model Uncertainty and Robust Duality in Finance

Bernt ØksendalCMA, University of Oslo, Norway

andNorwegian School of Economics (NHH),Bergen, Norway

Joint work with Agnès Sulem (INRIA Paris-Rocquencourt)

Conference on Stochastic Analysis and Applications,Hammamet, Tunisia, October 2013

IN N OVATION S IN S TOCHAS TIC AN AL Y S IS AN D AP P L ICAT ION Swith emphasis on S T OC H A S T I C C ON T R OL A N D I N F OR M A T I ON

Introduction

In the latest years, partly due to the financial crisis, there has beenan increased focus on model uncertainty in mathematical finance.Here model uncertainty is understood in the sense of uncertaintywith respect to the choice of the underlying probability measure.This is sometimes called Knightian uncertainty, after the Universityof Chicago economist Frank Knight (1885-1972). Thecorresponding model uncertainty stochastic control problem isoften called robust control.

Optimizing under model uncertainty is also of interest because it isrelated to risk minimization problems, where risk is interpreted inthe setting of a convex risk measure (Föllmer & Schied (2002),Frittelli & Rosazza-Gianin (2002)).


Outline

I Duality method in portfolio optimization; use of dualityrelationships in the space of convex functions andsemimartingales.

I Itô - Lévy market case

I Maximum principles for stochastic control

I Optimal portfolio and optimal scenario

I Model uncertainty and robust control

I Extension to robust duality

I Illustrating examples


Financial marketLet S(t) ; 0 ≤ t ≤ T represent discounted unit price of a riskyasset at time t. We assume that S(t) is a semimartingale on afiltered probability space (Ω,F , {Ft}t≥0,P).

T is a finite time horizon.

Let ϕ(t) be an Ft-predictable S-integrable portfolio process, givingthe number of units held of the risky asset at time t.If ϕ(t) is self-financing, the corresponding wealth processX (t) = X xϕ(t) is given by

X (t) = x +

∫ t0ϕ(s)dS(s) ; 0 ≤ t ≤ T , (0.1)

where x ≥ 0 is the initial value of the wealth.We say that ϕ is admissible and write ϕ ∈ A if Xϕ(t) ≥ 0 for allt ∈ [0,T ] a.s.


Let M be the set of probability measures Q which are equivalentlocal martingale measures (ELMM), in the sense that Q ∼ P andS(t) is a local martingale under Q.We assume that

M 6= ∅

which is related to absence of arbitrage opportunities on thesecurity market.


Let U : [0,∞]→ R be a utility function, strictly increasing, strictlyconcave, C1 and satisfying the Inada conditions:

U ′(0) = limx→0+

U ′(x) =∞ U ′(∞) = limx→∞

U ′(x) = 0.

Let V be the conjugate function of U:

V (y) := supx>0{U(x)− xy} ; y > 0

V is strictly convex, decreasing, C1 and satisfies

V ′(0) = −∞, V ′(∞) = 0,V (0) = U(∞), and V (∞) = U(0).

Moreover,

U(x) = infy>0{V (y) + xy} ; x > 0, and

U ′(x) = y ⇔ x = −V ′(y).

i.e. U ′ is the inverse function of −V ′.


Duality approach for optimal portfolio problemThe primal problem is to find ϕ∗ ∈ A such that

u(x) := supϕ∈A

E [U(X xϕ(T ))] = E [U(Xxϕ∗(T ))]. (0.2)

The dual problem to (0.2) is for given y > 0 to find Q∗ ∈M s.t.

v(y) = infQ∈M

E

[V

(ydQ

dP

)]= E

[V

(ydQ∗

dP

)]. (0.3)

Kramkov and Schachermayer (2003) prove that, under someconditions, ϕ∗ and Q∗ exist and are related by

U ′(X xϕ∗(T )) = ydQ∗

dPwith y = u′(x) (0.4)

i.e.

X xϕ∗(T ) = −V ′(ydQ∗

dP

)with x = −v ′(y). (0.5)

We extend this result by using stochastic control theory when therisky asset price S(t) is an Itô-Lévy process.


Itô-Lévy marketWe consider a financial market model, where the discounted unitprice S(t) of the risky asset is given by a jump diffusion of the formdS(t) = S(t−)

(btdt + σtdBt +

∫Rγ(t, ζ)Ñ(dt, dζ)

); 0 ≤ t ≤ T

S(0) > 0

(0.6)where bt , σt and γ(t, ζ) are predictable processes satisfyingγ(t, ζ) > −1 and

E

[∫ T0

{|bt |+ σ2t +

∫Rγ2(t, ζ)ν(dζ)

}dt

]

Primal ProblemLet ϕ(t) be a self-financing portfolio, giving the number of unitsheld of the risky asset at time t and let X (t) = X xϕ(t) be thecorresponding wealth process given by{dX (t) = ϕ(t)S(t−)

[btdt + σtdBt +

∫R γ(t, ζ)Ñ(dt, dζ)

]; 0 ≤ t ≤ T

X (0) = x > 0.

We say that ϕ is admissible and write ϕ ∈ A ifI ϕ(t) is an Ft-predictable S-integrable processI X (t) > 0 for all t ∈ [0,T ] a.s.I E

[∫ T0 ϕ(t)

2S(t)2{b2t + σ

2t +

∫R γ

2(t, ζ)ν(dζ)}dt] 0 s.t. E [∫ T

0 |X (t)|2+�dt]

Primal problem : Find ϕ∗ ∈ A such that

u(x) := supϕ∈A

E [U(X xϕ(T ))] = E [U(Xxϕ∗(T ))]. (0.8)


Dual ProblemWe represent the set M of ELMM by the family of positivemeasures Qθ of the form

dQθ(ω) = Gθ(T )dP(ω) on FT , where{dGθ(t) = Gθ(t

−)[θ0(t)dBt +

∫R θ1(t, ζ)Ñ(dt, dζ)

]; 0 ≤ t ≤ T

Gθ(0) = y > 0.

and θ = (θ0, θ1) is a predictable process satisfying the conditions

E

[∫ T0

{θ20(t) +

∫Rθ21(t, ζ)ν(dζ)

}dt

]

Let Θ denote the set of processes θ satisfying the above conditions.

Dual problem : For given y > 0 , find θ∗y ∈ Θ and v(y) s. t.

− v(y) := supθ∈Θ

E [−V (G yθ (T ))] = E [−V (Gθ∗y (T ))]. (0.10)

We will use the maximum principle for stochastic control to studythe primal problem and relate it to the dual problem.


Stochastic Control of Jump DiffusionsThere are two main methods for solving stochastic controlproblems:

(i) Dynamic programming and the HJB equation(R. Bellman ...) This method is very efficient when it works, but itassumes that the system is Markovian.

(ii) The maximum principle with the associated BSDE(Pontryagin, Bismut, Bensoussan, Pardoux-Peng, Framstad-Ø.-Sulem,...) This method is more robust; it does not assume thatthe system is Markovian and it applies even to partial information,to systems with delay, to FBSDEs and to SPDEs. The drawback isthat it involves complicated BSDEs, ABSDEs, BSPDEs, ...

Thus, in our general non-Markovian setting it is natural to use themaximum principle, and we will now use it to study the primal anddual problem above.


Maximum Principles for Stochastic Control

Consider the following general stochastic control problem:

supu∈A

E

[∫ T0

f (t,X (t), u(t), ω)dt + φ(X (T ), ω)

](0.11)

where A is a family of admissible F-predictable controls and

dX (t) = b(t,X (t), u(t), ω)dt + σ(t,X (t), u(t), ω)dBt (0.12)

+

∫Rγ(t,X (t), u(t), ω, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T ; X (0) = x ∈ R

Assume b, σ, γ, f , φ ∈ C 1 and

E [

∫ T0

(|∇b|2 + |∇σ|2 + ‖∇γ‖2 + |∇f |2)(t)dt]

For u ∈ A we assume the solution X u(t) of (0.12) exists, is uniqueand satisfies, for some � > 0, E [

∫ T0 |X

u(t)|2+�dt]

Theorem (Sufficient Maximum Principle, Ø. & Sulem, JOTA2012)

Let û ∈ A with corresponding solutions X̂ , p̂, q̂, r̂ be such that

supv∈U

H(t, X̂ (t), v , p̂(t), q̂(t), r̂(t, ·)) = H(t, X̂ (t), û(t), p̂(t), q̂(t), r̂(t, ·)).

Assume

I The function x 7→ φ(x) is concaveI (The Arrow condition) The function

H(x) := supv∈U

H(t, x , v , p̂(t), q̂(t), r̂(t, ·))

is concave for all t ∈ [0,T ].Then û is an optimal control for the problem (0.11).


Theorem (Necessary Maximum Principle, Ø. & Sulem, JOTA2012)

Assume

I For all t0 ∈ [0,T ] and all bounded Ft0-measurable randomvariables α(ω), the control β := 1[t0,T ]α belongs to A.

I For all u, β ∈ A with β bounded, there exists δ > 0 s.t.ũ := u + aβ belongs to A for all a ∈ (−δ, δ).

I The derivative process xt :=ddaX

u+aβ(t) |a=0, exists andbelongs to L2(dm × dP).

Then, if u∗ ∈ A is optimal,

∂H

∂u(t,X ∗(t), u∗, p∗(t), q∗(t), r∗(t, ·)) = 0 for all t ∈ [0,T ].


To illustrate the method, we first prove two useful auxiliary results:

LemmaPrimal problem.Let ϕ̂(t) ∈ A. Then ϕ̂(t) is optimal for the primal problem (0.8) ifand only if the (unique) solution (p̂1, q̂1, r̂1) of the BSDEdp̂1(t) = q̂1(t)dB(t) +

∫Rr̂1(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T

p̂1(T ) = U′(X xϕ̂(T )).

(0.14)satisfies the equation

b(t)p̂1(t) + σ(t)q̂1(t) +

∫Rγ(t, ζ)r̂1(t, ζ)ν(dζ) = 0 ; t ∈ [0,T ].

(0.15)


Proof. (Sketch)(i) First assume that ϕ̂ ∈ A is optimal for the primal problem(0.8). Then by the necessary maximum principle, thecorresponding Hamiltonian, given by

H1(t, x , ϕ, p, q, r) = ϕS(t−)(b(t)p+σq+

∫Rγ(t, ζ)r(ζ)Ñ(dt, dζ))

(0.16)satisfies

∂H1∂ϕ

(t, x , ϕ, p̂1(t), q̂1(t), r̂1(t, ·)) |ϕ=ϕ̂(t)= 0,

where (p̂1, q̂1, r̂1) satisfies (0.14) since∂H1∂x = 0. This implies

(0.15).


(ii) Conversely, suppose the solution (p̂1, q̂1, r̂1) of the BSDE(0.14) satisfies (0.15). Then ϕ̂, with the associated (p̂1, q̂1, r̂1)satisfies the conditions for the sufficient maximum principle, andhence ϕ̂ is optimal. �


We now turn to the dual problem (0.10). In the following weassume that

σ(t) 6= 0 for all t ∈ [0,T ]. (0.17)This is more because of convenience and notational simplicity thanof necessity.

LemmaDual problem.Let θ̂ ∈ Θ. Then θ̂ is an optimal scenario for the dual problem(0.10) if and only if the solution (p̂2, q̂2, r̂2) of the BSDEdp̂2(t) =

q̂2(t)

σ(t)b(t)dt + q̂2(t)dB(t) +

∫Rr̂2(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T

p̂2(T ) = −V ′(G yθ̂ (T )).(0.18)

also satisfies

− q̂2(t)σ(t)

γ(t, ζ) + r̂2(t, ζ) = 0 ; 0 ≤ t ≤ T . (0.19)


Proof.The Hamiltonian H2 associated to (0.10) is

H2(t, g , θ0, θ1, p, q, r) = gθ0q + g

∫Rθ1(ζ)r(ζ)ν(dζ). (0.20)

By (0.17), the constraint (0.9) can be written

θ0(t) = θ̃0(t) = −1

σ(t)

{b(t) +

∫Rγ(t, ζ)θ1(t, ζ)ν(dζ)

}; t ∈ [0,T ].

(0.21)


Substituting this into (0.20) we get

H̃2(t, g , θ1, p2, q2, r2) := H2(t, g , θ̃0, θ1, p2, q2, r2)

= g

(− q2σ(t)

{b(t) +

∫Rγ(t, ζ)θ1(ζ)ν(dζ)

}+

∫Rθ1(ζ)r2(ζ)ν(dζ)

).

(0.22)

The equation for the adjoint processes (p2, q2, r2) is thus thefollowing BSDE:

dp2(t) =

[q2(t)

σ(t)b(t) +

∫Rθ1(t, ζ)

(q2(t)

σ(t)γ(t, ζ)− r2(t, ζ)

)ν(dζ)

]dt

+q2(t)dB(t) +

∫Rr2(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T

p2(T ) = −V ′(Gθ(T )).(0.23)


If there exists a maximiser θ̂1 for H̃2 then

(∇θ1H̃2)θ1=θ̂1 = 0, (0.24)

i.e.

− q̂2(t)σ(t)

γ(t, ζ) + r̂2(t, ζ) = 0 ; 0 ≤ t ≤ T , (0.25)

where (p̂2, q̂2, r̂2) is the solution of (0.23) corresponding to θ = θ̂.We thus get (0.18) and this ends the necessary part.

The sufficient part follows from the fact that the functionsg → −V (g) and

g → supθ1

H̃2(t, g , θ1, p̂2(t), q̂2(t), r̂2(t, ·)) = −gq̂2(t)

σ(t)b(t)

are concave. �


Optimal Scenario and Optimal Portfolio

We now use the above general machinery of stochastic control toobtain an explicit connection between an optimal portfolio ϕ̂ ∈ Afor the primal problem and an optimal θ̂ ∈ Θ for the dual problem:


Theorem 1 a) Suppose ϕ̂ ∈ A is optimal for the primal problem

supϕ∈A

E [U(X xϕ(T ))].

Let (p1, q1, r1) be the associated adjoint processes, solution of theBSDEdp1(t) = q1(t)dB(t) +

∫Rr1(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T

p1(T ) = U′(X xϕ̂(T )).

Define

θ̂0(t) :=q1(t)

p1(t−); θ̂1(t, ζ) :=

r1(t, ζ)

p1(t−).

Suppose E [∫ T

0 {θ̂20(t) +

∫R θ̂

21(t, ζ)ν(dζ)}dt] −1.


Then θ̂ = (θ̂0, θ̂1) ∈ Θ is optimal for the dual problem

supθ∈Θ

E [−V (G yθ (T ))]

with initial value y = p1(0) .

Moreover, the optimal density process G yθ̂

starting at y = p1(0)coincides with the optimal adjoint process p1 for the primalproblem, i.e.

G yθ̂

(t) = p1(t); 0 ≤ t ≤ T . (0.26)

In particular,G yθ̂

(T ) = U ′(X xϕ̂(T )). (0.27)


b) Conversely, suppose θ̂ = (θ̂0, θ̂1) ∈ Θ is optimal for the dualproblem

supθ∈Θ

E [−V (G yθ (T ))]

Let (p2, q2, r2) be the associated adjoint processes, solution of theBSDEdp2(t) =

q2(t)

σ(t)b(t)dt + q2(t)dBt +

∫Rr2(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T

p2(T ) = −V ′(G yθ̂ (T )).

Then

ϕ̂(t) :=q2(t)

σtS(t−)1σt 6=0 +

r2(t, ζ)

γ(t, ζ)S(t−)1σt=0,γ(t,ζ)6=0

is an optimal portfolio (if it is admissible) for the primal problem

supϕ∈A

E [U(X xϕ(T ))]

with initial value x = p2(0).


Moreover, the optimal wealth process X xϕ̂ starting at x = p2(0)coincides with the optimal adjoint process p2 for the dual problem,i.e.

X xϕ̂(t) = p2(t); 0 ≤ t ≤ T . (0.28)

In particularX xϕ̂(T ) = −V ′(G

y

θ̂(T )). (0.29)


ExampleNo jumps (N = 0). Then Θ has just one element θ̂ given by

θ̂(t) = −btσt.

and

G yθ̂

(T ) = y exp(−∫ T

0

bsσs

dBs −1

2

∫ T0

b2sσ2s

ds).

Therefore, by Theorem 2 b), if (p2, q2) is the solution of the BSDEdp2(t) =q2(t)

σtbtdt + q2(t)dBt ; 0 ≤ t ≤ T

p2(T ) = −V ′(Gθ̂(T )),(0.30)

then ϕ̂(t) :=q2(t)

σ(t)S(t−)is an optimal portfolio for the problem

supϕ∈A

E [U(X xϕ(T ))]

with initial value x = p2(0).


In particular, if U(x) = ln x , then V (y) = − ln y − 1 andV ′(y) = −1

y. So the BSDE (0.30) becomes

dp2(t) =q2(t)

σtbtdt + q2(t)dBt ; 0 ≤ t ≤ T

p2(T ) =1

yexp

(∫ T0

bsσs

dBs +1

2

∫ T0

b2sσ2s

ds

).

(0.31)

To solve this equation we try q2(t) = p2(t)btσt. Then

dp2(t) = p2(t)

[b2tσ2t

dt +btσt

dBt

], (0.32)

which has the solution

p2(t) =1

yexp

(∫ t0

bsσs

dBs +1

2

∫ t0

b2sσ2s

ds

); 0 ≤ t ≤ T . (0.33)

Hence (0.31) holds.


We conclude that the optimal portfolio for primal problem withinitial value x = 1y is

ϕ̂(t) = p2(t)bt

σ2t S(t−). (0.34)

Note that with this portfolio we get

dXϕ̂(t) = p2(t)btσ2t

[btdt + σtdBt ]

= p2(t)

[b2tσ2t

dt +btσt

dBt

]= dp2(t). (0.35)


Therefore

ϕ̂(t) = Xϕ̂(t)bt

σ2t S(t−)

(0.36)

which means that the optimal fraction of wealth to be placed inthe risky asset is

π̂(t) =ϕ̂(t)S(t−)

Xϕ̂(t)=

btσ2t, (0.37)

which agrees with the classical result of Merton.


Model uncertainty setupFor a different approach to robust duality see Gushkin (2011)[2].See also the survey in Föllmer et al (2009) [1].

To get a representation of model uncertainty, we consider a familyof probability measures R = Rκ ∼ P, with Radon-Nikodymderivative on Ft given by

d(Rκ | Ft)d(P | Ft)

= Zκt

where, for 0 ≤ t ≤ T , Zκt is a martingale of the form

dZκt = Zκt− [κ0(t)dBt +

∫Rκ1(t, ζ)Ñ(dt, dζ)] ; Z

κ0 = 1.

Let K denote a given set of admissible scenario controlsκ = (κ0, κ1), Ft-predictable, s.t. κ1(t, z) ≥ −1 + �, andE [∫ T

0 {|κ20(t)|+

∫R κ

21(t, z)ν(dz)}dt]

By the Girsanov theorem, using the measure Rκ in stead of theoriginal measure P in the computations involving the price processS(t), is equivalent to using the original measure P in thecomputations involving the perturbed price process Sµ(t) in steadof P(t), where Sµ(t) is given by

dSµ(t) = Sµ(t−)[(bt + µtσt)dt + σtdBt +∫Rγ(t, ζ)Ñ(dt, dζ)]

Sµ(0) > 0,

(0.38)with

µtσt = −σtκ0(t)−∫Rγ(t, ζ)κ1(t, ζ)ν(dζ)dt (0.39)


Recall that a measure Rκ is an Equivalent Local MartingaleMeasure (ELMM) iff κ = (κ0, κ1) are such that

bt + σtκ0(t) +

∫Rγt(ζ)κ1(t, ζ)ν(dζ) = 0.

Note that we do not assume a priori that the measures Rκ for κ ∈K are ELMMs.


Robust Stochastic Control

Accordingly, we now replace the price process S(t) by theperturbed processdSµ(t) = Sµ(t−)[(bt + µtσt)dt + σtdBt +

∫Rγ(t, ζ)Ñ(dt, dζ)]

Sµ(0) > 0,

(0.40)for some predictable perturbation process µt , assumed to satisfy

E[∫ T

0 |µtσt |dt] 0.


Let A denote the set of portfolios ϕt such that

E

[∫ T0ϕ2tSµ(t)

2

{(bt + µtσt)

2 + σ2t +

∫Rγ2(t, ζ)ν(dζ)

}dt

] 0 for all t ∈ [0,T ], a.s.

and ∃� > 0 s.t.

E [

∫ T0|X (t)|2+�dt]

Let ρ : R→ R be a convex penalty function, assumed to be C1.The robust primal problem is to find (ϕ̂, µ̂) ∈ A×M such that

infµ∈M

supϕ∈A

I (ϕ, µ) = I (ϕ̂, µ̂) = supϕ∈A

infµ∈M

I (ϕ, µ), (0.41)

where

I (ϕ, µ) = E

[U(Xϕ,µ(T )) +

∫ T0ρ(µt)dt

]. (0.42)

This a stochastic differential game that we handle by usingmaximum principle. Define the Hamiltonian by

H1(t, x , ϕ, µ, p, q, r) = ρ(µ)

+ ϕSµ(t−)

[(bt + µσt)p + σtq +

∫Rγ(t, ζ)r(ζ)ν(dζ)

].

Since ∂H1∂x = 0, the BSDE for the adjoint processes (p1, q1, r1) is

dp1(t) = q1(t)dBt +

∫Rr1(t, ζ)Ñ(dt, dζ); p1(T ) = U

′(Xϕ,µ(T )).


First order conditions for a maximum ϕ̂ and a minimum µ̂:

(bt + µ̂tσt)p1(t) + σtq1(t) +

∫Rγ(t, ζ)r1(t, ζ)ν(dζ) = 0 (0.43)

ρ′(µ̂t) + ϕ̂tσtp1(t)Sµ(t−) = 0 (0.44)

Since H1 is concave with respect to ϕ and convex with respect toµ, these first order conditions are also sufficient.Therefore we obtain the following characterization of a saddlepoint of (0.41):

Theorem 2: A pair (ϕ̂, µ̂) ∈ A×M is a solution of the robustprimal game problem (0.41) iff the solution (p1, q1, r1) of the BSDEdp1(t) = q1(t)dBt +

∫Rr1(t, ζ)Ñ(dt, dζ) ; 0 ≤ t ≤ T

p1(T ) = U′(Xϕ̂,µ̂(T )).

(0.45)satisfies (0.43), (0.44).


The robust dual problem is to find θ̂ ∈ Θ, µ̂ ∈M such that

supµ∈M

supθ∈Θ

J(θ, µ) = J(θ̂, µ̂) = supθ∈Θ

supµ∈M

J(θ, µ) (0.46)

where

J(θ, µ) = E

[−V (Gθ(T ))−

∫ T0ρ(µ(t))dt

], (0.47)

V is the conjugate function of U and G (t) = Gθ,µ(t) is given bydG (t) = G (t−)[θ0(t)dBt +

∫Rθ1(t, ζ)Ñ(dt, dζ)

]; 0 ≤ t ≤ T

G (0) = y > 0,

(0.48)with the constraint that if y = 1, then the measure Qθ,µ defined by

dQθ,µ = G (T )dP on FT

is an ELMM for the perturbed price process Sµ(t).


By the Girsanov theorem, this is equivalent to requiring that(θ0, θ1) satisfies

bt + µtσt + σtθ0(t) +

∫Rγ(t, ζ)θ1(t, ζ)ν(dζ) = 0 (0.49)


Substituting

θ0(t) = −1

σt

[bt + µtσt +

∫Rγ(t, ζ)θ1(t, ζ)ν(d , ζ)

](0.50)

into (0.48) we getdG (t) = G (t−)

(− 1σt

[bt + µtσt +

∫Rγ(t, ζ)θ1(t, ζ)ν(dζ)

]dBt

+

∫Rθ1(t, ζ)Ñ(dt, dζ)

); 0 ≤ t ≤ T

G (0) = y > 0.

(0.51)The Hamiltonian becomes

H2(t, g , θ1, µ, p, q, r) = −ρ(µ)−gq

σt

[bt + µσt +

∫Rγ(t, ζ)θ1(ζ)ν(dζ)

]+ g

∫Rθ1(ζ)r(ζ)ν(dζ). (0.52)


The BSDE for the adjoint processes (p2, q2, r2) is

dp2(t) = (q2(t)

σt[bt + µtσt +

∫Rγt(ζ)θ1(t, ζ)ν(dζ)]−

∫Rθ1(t, ζ)r2(t, ζ)ν(dζ))dt

+ q2(t)dBt +

∫Rr2(t, ζ)Ñ(dt, dζ) ; p2(T ) = −V ′(G (T )).

(0.53)

The first order conditions for a maximum point (θ̃, µ̃) for H2 are

(∇θ1H2 =)−q2(t)

σtγt(ζ) + r2(t, ζ) = 0 (0.54)(

∂H2∂µ

=

)ρ′(µ̃t) + Gθ̃,µ̃(t)q2(t) = 0. (0.55)


Substituting (0.54) into (0.53) we getdp2(t) =q2(t)

σt[b2(t) + µ̃tσt ]dt + q2(t)dBt +

∫Rr2(t, ζ)Ñ(dt, dζ) ; t ∈ [0,T ]

p2(T ) = −V ′(Gθ̃,µ̃(T )).(0.56)

Theorem 4: A pair (θ̃, µ̃) ∈ Θ×M is a solution of the robust dualproblem (0.46)-(0.47) if and only the solution (p2, q2, r2) of theBSDE (0.56) also satisfies (0.54)-(0.55), i.e.

−q2(t)σt

γt(ζ) + r2(t, ζ) = 0

ρ′(µ̃t) + Gθ̃,µ̃(t)q2(t) = 0


From robust primal to robust dual

We now use the characterizations above of the solutions(ϕ̂, µ̂) ∈ A×M and (θ̃, µ̃) ∈ Θ×M of the robust primal and dualproblems, to find the relations between them.


Theorem : Assume (ϕ̂, µ̂) ∈ A×M is a solution of the robustprimal problem and let (p1, q1, r1) be the associated adjointprocesses. Then,

µ̃ = µ̂ (0.57)

θ̃0(t) =q1(t)

p1(t−); θ̃1(t, ζ) =

r1(t, ζ)

p1(t−). (0.58)

are optimal for dual problem with initial value y = p1(0).Moreover the optimal adjoint process p1 for the robust primalproblem coincides with the optimal density process Gθ̃,µ̃ for therobust dual problem, i.e.

p1(t) = Gθ̃,µ̃(t); 0 ≤ t ≤ T . (0.59)

In particular,U ′(Xϕ̂,µ̂(T )) = Gθ̃,µ̃(T ). (0.60)


From robust dual to robust primalTheorem :Let (θ̃, µ̃) ∈ Θ×M be optimal for the robust dual problem and let(p2, q2, r2) be the associated adjoint processes. Then

µ̂ := µ̃

ϕ̂t :=q2(t)

σtS(t−)1σt 6=0 +

r2(t, ζ)

γ(t, ζ)S(t−)1σt=0,γ(t,ζ) 6=0 ; t ∈ [0,T ].

are optimal for primal problem with initial value x = p2(0).Moreover, the optimal adjoint process p2 for the robust dualproblem coincides with the optimal state process Xϕ̂,µ̂ for therobust primal problem, i.e.

p2(t) = Xϕ̂,µ̂, 0 ≤ t ≤ T (0.61)

In particular− V ′(Gθ̃,µ̃(T )) = Xϕ̂,µ̂(T ). (0.62)


Example

We study the robust primal problem

infµ∈M

supϕ∈A

E

[U(Xϕ,µ(T )) +

∫ T0ρ(µt)dt

]. (0.63)

with no jumps.Then there is only one ELMM for the price processSµ, and the corresponding robust dual problem simplifies to

supµ∈M

E

[−V (Gµ(T ))−

∫ T0ρ(µt)dt

], where (0.64)

dGµ(t) = −Gµ(t−)[btσt

+ µt ]dBt ; Gµ(0) = y > 0 (0.65)


The first order conditions for the Hamiltonian reduce to:

µ̃t = ρ′−1(−Gµ̃(t)q2(t)) (0.66)

which substituted into the adjoint BSDE gives:dp2(t) = q2(t)[btσt

+ ρ′−1

(−Gµ̃(t)q2(t))]dt + q2(t)dBt ; t ∈ [0,T ]

p2(T ) = −V ′(Gµ̃(T )).(0.67)


Now assume that

U(x) = ln x and ρ(µ) =1

2µ2. (0.68)

Then V (y) = − ln y − 1.If bt and σt are deterministic, we can use dynamic programmingand this leads to the solution

µ̃t = −bt

2σt; t ∈ [0,T ] (0.69)

We now check that this also holds if bt and σt are Ft-adaptedprocesses, by verifying that the system (0.65)-(0.67) is consistent:


This system is now

Gµ̃(t) = y exp(−∫ t

0

bs2σs

dBs −1

2(bs

2σs)2ds) (0.70)

q2(t) =1

Gµ̃(t).bt

2σt(0.71)

dp2(t) =1

Gµ̃(t)

[bt

2σtdBt + (

bt2σt

)2dt

]; p2(T ) =

1

Gµ̃(T )(0.72)

which gives

d(1

Gµ̃(t)) =

1

Gµ̃(t)

[bt

2σtdBt + (

bt2σt

)2dt

]. (0.73)


We see that (0.72) is in agreement with (0.73), so

µ̃t = −bt

2σt

is indeed optimal.The corresponding optimal portfolio for therobust primal is

ϕ̂t =bt

Gµ̃(t)2σtSµ̃(t−); t ∈ [0,T ]. (0.74)


We have proved:

TheoremSuppose (0.68) hold. Then the optimal scenario µ̂ = µ̃ andoptimal ϕ̂ for the robust primal problem (0.63) are given by

µ̃t = −bt

2σt; t ∈ [0,T ] (0.75)

and

ϕ̂t =bt

Gµ̃(t)2σtSµ̃(t−); t ∈ [0,T ]. (0.76)

respectively, with Gµ̃(t) as in (0.70).

It is interesting to compare this result with the solution we found inthe corresponding previous example in the non-robust case (ρ = 0):

ϕ̂(t) = Xϕ̂(t)bt

σ2t S(t−)

(0.77)


Summary

I The purpose of this presentation has been to use stochasticcontrol theory to obtain new results and new proofs of resultsin portfolio optimization both with and without modeluncertainty. In the case with no model uncertainty, then someof the results have been proved earlier by using convex dualitytheory.

I The advantage of this approach is that it gives an explicitrelation between the optimal measure in the dual formulationand the optimal portfolio in the primal formulation, both inthe cases with and without model uncertainty. BSDEs play acrucial role in this connection.


Föllmer, H., Schied, A., Weber, S.: Robust preferences androbust portfolio choice, In: Mathematical Modelling andNumerical Methods in Finance. In: Ciarlet, P., Bensoussan, A.,Zhang, Q. (eds): Handbook of Numerical Analysis 15, pp.29-88 (2009)

Gushkin, A. A.: Dual characterization of the value function inthe robust utility maximization problem. Theory Probab. Appl.55 (2011), 611-630.

Hu, Y. and Peng, S.: Solution of forward-backward stochasticdifferential equations, Probab.Theory Rel. Fields (103),273-283 (1995).

Kramkov, D. and Schachermayer, W.: Necessary and sufficientconditions in the problem of optimal investment in incompletemarkets. Ann. Appl. Probab. 13 (2003), 1504-1516.


Øksendal, B., Sulem, A.: Forward-backward stochasticdifferential games and stochastic control under modeluncertainty. J. Optim. Theory Appl.(2012).

Quenez, M.C., Sulem, A.: BSDEs with jumps, optimizationand applications to dynamic risk measures. Stoch. Proc. andtheir Appl. (2013)


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