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Stochastic Differential Equations SDE

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:.

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Stochastic Differential Equations SDE

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  1. Stochastic Differential EquationsSDE Peyman Givi Department of Mechanical Engineering and Materials Science University of Pittsburgh October, 2009

  2. Objective To predict and understand of Stochastic Process, or sometimes Random Process. • Numerical solution

  3. 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:

  5. Random Process: • Weak:

  6. Random Process: • Strong:

  7. Gaussian

  8. Random Process: • Diffusion Process: consider the stochastic Process which governed by: • If Wiener process Diffusion coefficient is a deterministic ODE that can be solved by classical calculus

  9. 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 famouschapman-kolmogorov relation which results in Foker-Plank equation: PDF of z(t) for given

  10. Multi-Dimensional Diffusion Process • Where Vector Matrix Vector

  11. Numerical Solution of SDE: • Lets consider 1-D • Euler method: Gaussian random number with zero mean and variance of That’s why we can not use classical calculus

  12. ItÔ Rule: • Consider diffusion Process • ItÔ Rule : consider G as any random process which is a function of the z • After substitution: ItÔ Rule

  13. 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

  14. LMSE: • Consider LMSE as: • Mean: • Moments:

  15. LMSE: We can verify all of these results numerically for various initial conditions Recommend: Paper of Kosaly and Givi

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

  17. THANK YOU!

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