16 janvier 2014 : Hernán Mardones (Concepción)

Numerical methods for stochastic differential equations

This talk addresses the numerical approximation of E f(X_{t}), where f : R^{d} → R and (X_{t})_{t≥0} solves the d-dimensional autonomous Itô stochastic differential equation (SDE) :

,

with b, σ^{1}, ..., σ^{m} : R^{d} → R^{d}, W_{1}, ..., W_{m} independent Wiener processes defined on a complete probability space (Ω, F, (F_{t})_{t≥0}, P), and X_{0} a random variable independent of W_{1}, ..., W_{m}.

In many areas, SDEs arise in modelling as a more realistic representation of physical systems by incorporating random effects. Most of the time, there is no analytical solution known but the availability of computational tools allows us to use numerical techniques.

Stochastic models may be hard to solve numerically. For example, in the case of multiplicative noise, the use of explicit methods (like the Euler-Maruyama scheme) can lead to poor approximations if the time step is not small enough. Usually implicit methods (e.g. the backward Euler method) are more convenient, but at the price of additional computational effort, because often they require the solution of an algebraic equation at each time step.

Notions of convergence with respect to smooth test functionals and stability properties of numerical methods will be discussed. In the scalar context, some criteria for the almost-sure exponential stability and the sign-preserving property will be presented. For the multidimensional case, we will concentrate on the design of stable schemes for bilinear systems of SDEs. Finally, some numerical experiments comparing the performance of Euler-type schemes with drift-implicit newer ones will be considered to illustrate the theory.