Lecture I : Markov chains
Theory : transition probability and Markov property,
classification of states, ergodic theorems, entrance and exit times,
generalization to continuous-in-time Markov chains, random walks.

Numerics: random number generator, Box-Mueller, application to
simulation of Markov chain, reconstruction of continuum Markov chain
from discrete sampling.

Lecture II : Wiener process
Theory invariance principle, elementary properties,
alternative constructions, general stochastic processes: definition
and properties.

Numerics: simulation of Brownian path, representation by Karhunen-Loeve
expansion, Haar basis representation.

Lecture III: Stochastic differential equations
Theory : Ito integrals and Ito
differential equations, Ito isometries, Ito formula, Stratonovich SDEs.

Numerics: Rieman sum for stochastic integrals, Euler-Maruyama method,
strong and weak convergence, stochastic Taylor expansion and higher
order schemes, integration over long time intervals.

Lecture IV: Fokker Planck equations
Theory: derivation, applications, examples.

Numerics: standard numerical schemes, boundary conditions, solution via
SDE simulation, Monte Carlo methods.

Lecture V: Path integral representation
Theory: Wiener measure, Cameron-Martin
formula, Girsanov formula, Feyman-Kac formula, application to the
theory of rare events.

Numerics: Monte-Carlo in path-space, minimum action method.

GUEST LECTURER

Ibrahim Fatkullin
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Stochastic models for tropical convection and climate

Claude Le Bris
Polymeric fluids

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Data Analysis of Time Series: A Case Study