Stochastic Integration and Stochastic Differential Equations a Gentle Introduction

7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction

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Intro Brownian motion Integration SDE Applications

Stochastic Integration and Stochastic DifferentialEquations: a gentle introduction

Oleg MakhninNew Mexico Tech

Dept. of Mathematics

October 26, 2007

Oleg Makhnin New Mexico Tech Dept. of Mathematics

Stochastic Integration and Stochastic Differential Equations: a gentle introduction


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Intro: why Stochastic?

Brownian Motion/ Wiener process

Stochastic IntegrationStochastic DEs

Applications




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Intro

Deterministic ODE X(t) = a(t,X(t)) t> 0X(0) = X0

However, noise is usually presentX(t) = a(t,X(t)) + b(t,X(t))(t), t> 0X(0) = X0

Natural requirements for noise :

- (t) is random with mean 0- (t) is independent of(s), t= s White noise- is continuous

Strictly speaking, does not exist!




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Brownian motion

T

c

T

T

c

$$$

Intuitive: Random walk, DiffusionStocks: you cannot predict the future behavior based on pastperformance




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Brownian motion

0 200 400 600 800 1000

60

40

20

0

20

40

60

Several paths of Brownian Motion and LIL bounds

t

Bt



S


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2D Brownian motion



I B i i I i SDE A li i


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Brownian motion: Axioms

BM is continuous (but not smooth, see later!), W(0) = 0

Normality:W(t+t)W(t) Normal(mean = 0, variance 2 = t)

3 2 1 0 1 2 3

0

.0

0.

1

0.

2

0.

3

0.

4

2

=

t

Normal because sum of many small, independent increments.



I t B i ti I t ti SDE A li ti


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Brownian motion: Axioms

Independence:

0.00 0.02 0.04 0.06 0.08 0.10

0

.4

0.3

0.2

0.1

0.0

t

W(t)

0.00

0

.4

0.3

0.2

0.1

0.0

W(t)





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Fractal nature

0.0 0.2 0.4 0.6 0.8 1.0

0 1 2 3 4 5

0 5 10 15 20 25

0.0 0.2 0.4

0 1 2

0 5 10





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Integration

Need

T

0 h(t) dW(t)

How would you define it?- For which functions h?- Answer is random (since W(t) is)

Riemann integral

T

0h(t) dt

ni=1

h(ti )ti

Stieltjes integral T

0

h(t) dF(t)

n

i=1

h(ti )[F(ti+1)

F(ti)]

where F has finite variation

- when F exists, then

T

0h(t) dF(t) =

T

0h(t)F(t) dt





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Stochastic Integration

Try the same:

T

0h(t) dW(t)

ni=1

h(ti )[W(ti+1)W(ti)]

does this still work?

No, W does not have finite variation

let maxti 0, limit in what sense? Depends on ti

Iton

i=1

h(ti)[W(ti+1)W(ti)]Stratonovichn

i=1

h

ti + ti+1

2

[W(ti+1)W(ti)]





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Stochastic Integration: examples

T

0dW(t)

W(ti+1)W(ti) = W(T) of course (recall W(0) = 0)

T

0t dW(t)

ti[W(ti+1)W(ti)] =? to be continued

so far, Ito and Stratonovich integrals agree.However,

T

0W(t) dW(t)

W(ti)[W(ti+1)W(ti)]





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g pp

Stochastic Integration:T

0 W(t) dW(t)

note,2

W(ti)[W(ti+1)W(ti)] = W2(ti+1)W

2(ti) (W(ti+1)W(ti))

2

W2(T) T(brown part by Law of Large numbers)

Hence,

T

0W(t) dW(t) =

W2(T)

2 T

2

Wish it were easier?





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Stochastic Differential Equations

X(t) =

t

0b(s,X(s)) dW(s), t> 0

dX(s) = b(s,X(s)) dW(s) - Stochastic differentialFull form

dX(s) = a(s,X(s)) ds + b(s,X(s)) dW(s),X(0) = X0 (can be random)

(note that dW/dt does not exist)





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Ito Formula

Let dX(t) = a(t,X(t)) dt + b(t,X(t)) dW(t)

then, the chain rule is

dg(t,X(t))(t) = gtdt+gxdX(t) + gxxb2(t,X(t))

2dt - extra term

heuristic: follows from Taylor series assuming (dW(t))2

= dt





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for example,

d(W2(s)) = W2(s + ds)W2(s)= [W(s + ds) + W(s)][W(s + ds)W(s)]= [W(s + ds)

W(s)][W(s + ds)

W(s)]

+2W(s)[W(s + ds)W(s)]= 2W(s) dW(s) + ds

Hence,

t

0 W(s) dW(s) =

W2(t)

2 t

2

Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction



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dg(t,X(t)) = gtdt+ gxdX(t) + gxxb2

2 dt

Exercise:

d(W3(s)) =?

= 3W2(s) dW(s) + 3W(s)ds

therefore,t

0W2(s) dW(s) = W

3

(t)3

t0

W(s) ds

d(sW(s)) =?

= W(s)ds + s dW(s),

therefore

t

0s dW(s) = tW(t)

t

0W(s) ds




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Solution

What does the solution look like?

Note that for Ito integrals, we always have

Et

0

h(s) dW(s) = 0 Expected value

E

t

0h(s) dW(s)

2=

t

0E[h2(s)] ds Variance

(can see using the Riemann sums).For example, the result of

t

0s dW(s) is a Normal random

variable, with mean 0 and variance = t3/3




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Langevin equation

More realistic Brownian motion: resistance/friction

dX(t) = X(t)dt + dW(t)

X(t) = particle velocityThen, can obtain using Ito formula, integrating factor et

X(t) = etX0 + t

0

e(ts) dW(s)




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Stock price

dX(t) = rX(t) dt + X(t) dW(t)

can express d log(X(t)) and get

X(t) = exp

(r 22

)t + W(t)

Note that for both equations, expected value E[X(t)] coincideswith the solution of deterministic equation, here

d X(t) = r X(t) dt




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A nonlinear equation

Consider a logistic equation for population dynamics: non-linear

dX(t) = rX(t)(1

X(t)) dt + X(t) dW(t)

Here, deterministic solution of

d X(t) = r X(t)(1 X(t)) dt

is different from the mean of stochastic solutions




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A nonlinear equation

dX(

t) =

rX(

t)(1

X(

t))

dt+

X(

t)

dW(

t)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

t

X(t)




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Math Finance




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Math Finance

- Optimal control of investments

- Black-Scholes formula for option pricing

- Based ondX(t) = rX(t) dt + X(t) dW(t)

and beyond




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Other applications

Boundary value problems: e.g. solving a Dirichlet problemf = 0 (harmonic) in the region U Rnf = g on the boundary U, given g

a stochastic solution is

f(x) = Ex[g(point of first exit from U)],

Ex refers to Brownian motion started from x.




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Other applications

Hydrology: porous media flow

div(V) = 0 incompressible flowV =

K

P Darcys law

V = velocity, P = pressure fieldK = conductivity is stochastic

Filtering: determine the position of a stochastic process from noisyobservation history. Kalman filter (discrete), Kalman-Bucy

filter (continuous)

Predator-prey models, Chemical reactions, Reservoir models ... ...




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QUESTIONS?




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THANK YOU!


Stochastic Integration and Stochastic Differential Equations a Gentle Introduction

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