Numerical methods for SDEs, contents

Stochastic integrals. The Ito formula.
Stochastic differential equations. Existence and uniqueness. The Stratonovich integral.
Connections to PDE:s. The Kolmogorov equations. Feynman-Kac's formula.

Ito integrals is the limit of forward Euler; examples and a short proof for Lipschitz functions.
Kolmogorov's backward and forward equations and relations to Monte Carlo methods. Simple proof of Feynman-Kac's formula. Discussion on PDE versus Monte Carlo methods.
Approximations of PDE; finite difference, FEM. Variational inequalities for American Options.

Introduction to SDEs: models, different noise processes, stochastic integrals, Taylor series, expectations.
Numerical methods and their order properties: weak and strong order, stochastic Runge-Kutta methods, stochastic linear multistep methods, difficulties with lack of commutativity in the problem, the Magnus formula, numerical results, B-series and convergence of methods.
Stability properties and implicit methods: A-stability, MS-stability, T-stability, stiffness, composite methods, implicit methods.
An application in hydrology - the numerical solution of a stochastic partial differential equation: Wiener processes in time and space computation techniques.
Implementation issues: computation of stochastic integrals, the Brownian path, variable step size implementations, embedding, extrapolation, PI control.

Introduction: review of state-space representations of SDEs, basic background on the evolution of probability density functions for systems of SDEs, some examples to help motivate and visualize PDF evolution.
The Fokker-Planck equation: what is it, where does it come from, why do we care about it, how do we use it.
Discretization of the Fokker-Planck equation: spatial discretization using finite element methods, the finite element method, application to the F-P eqn, spatial discretization using finite difference methods, time discretization,
Solution methods.
Visualization of PDF evolutions.
Examples and applications.