University of Pittsburgh

Stochastic Differential EquationsSDEPeyman Givi

Department of Mechanical Engineering and Materials Science

University of PittsburghOctober, 2009

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ObjectiveTo predict and understand of Stochastic Process, or sometimes Random Process.

Numerical solution

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Random Process:•Random walk: the motion of a drunk person with possibility of ½ forward or ½ backward.

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Random Process:•Wiener Process: the random walk in the limit of weak:

Strong: this is the most utilized means of constructing wiener process:

0t

wtwtw )1()(

21Pr

obttw

wtwtw )1()(

)1,0(1)(

0)()1(

2N

E

Ettw

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Random Process:•Weak:

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Random Process:•Strong:

Gaussian

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Random Process:•Diffusion Process: consider the stochastic Process which governed by:

•If

)(]),([]),([)( tdwttzdtttzmtdz

Diffusion coefficient Wiener process0

]),([)(]),([)(

ttzmdttdz

dtttzmtdz is a deterministic ODE that can be solved by classical calculus

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Random Process:•Foker-Planck equation of diffusion Process: for stochastic process, governed by diffusion process there is a transitional PDF:

•The evolution of P is derived via the use of famous chapman-kolmogorov relation which results in Foker-Plank equation:

],),([ 00 tzttzPPDF of z(t) for given

]),([21]),([ 2

2

2

Ptzz

Ptzmzt

P

00 )( ztz

Multi-Dimensional Diffusion Process

)(]),([]),([)( twdttzdtttzmtzd

r

rn

n

tw

m

)(

•Where VectorMatrixVector

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Numerical Solution of SDE:•Lets consider 1-D

•Euler method:

dwtzdttzmtdz ),(),()(

tw

twttttzmtZttZ

)()()),(()()(Gaussian random number with zero mean and variance of t

tttttzmtZttZ )()),(()()(

That’s why we can not use classical calculus

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ItÔ Rule:•Consider diffusion Process

•ItÔ Rule : consider G as any random process which is a function of the z

•After substitution:

dwttzdtttzmtdz )),(()),(()(

...)(21...)(

21

]),([

22

22

2

2

dzzGdz

zGdt

tGdt

tGdG

ttzG

dwzGdt

zG

zGm

tGdG

2

22

2

ItÔ Rule

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Monte Carlo Methods• M/C Method: are a class of computational algorithms

that rely on repeated random sampling to compute their results. Monte Carlo methods are often used when simulating physical and mathematical systems. Because of their reliance on repeated computation of random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. Monte Carlo methods tend to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithm.

• Example: Calculation of 1415.3

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LMSE:•Consider LMSE as:

•Mean:

•Moments:

dtPt

tPP

PtP

),()(

),(

)(

mm

mm

mdt

d

dPdtP

dtd

dPdtP

)()()(

0

)(

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LMSE:

We can verify all of these results numerically for various initial conditionsRecommend: Paper of Kosaly and Givi

wt

wt

wt

mm

etdtd

etdtd

etdtd

mdt

d

4443

444

3333

333

2222

222

)0()(4

)0()(3

)0()(2

)0()()()(

)0()(

)()(

422

44

323

2

33

ttt

t

tt

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Numerical exercises

There are plenty of useful exercises on notes for better understanding the concept of stochastic process.

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THANK YOU!