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Stochastic Differential Equations SDE. Peyman Givi Department of Mechanical Engineering and Mater ials Science University of Pittsburgh October, 2009. Objective. To predict and understand of Stochastic Process, or sometimes Random Process. Numerical solution . Random Process:. - PowerPoint PPT Presentation
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University of Pittsburgh
Stochastic Differential EquationsSDEPeyman Givi
Department of Mechanical Engineering and Materials Science
University of PittsburghOctober, 2009
2
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.
4
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!