MATHEMATICS COLLOQUIUM

 WHEN:
On Thursday, February 03, 2000 at 2:20 PM

WHERE:
Durham Science Center, Room 255

WHAT:

Henri Schurz
School of Mathematics
University of Minnesota, Minneapolis

will give a talk on

Stochastic Differential Equations, Numerical Analysis and Applications

Abstract:
During the talk we will sketch some basic aspects on stochastic dynamical systems, in particular on the prototypes of stochastic differential equations (SDEs), their stochastic-numerical methods and some potential applications. These objects are often met in Natural and Environmental Sciences, Financial Markets and Marketing when ``uncertainty'' in modeling or ``erratic behavior'' of the environment plays an essential role.

The formulas of Dynkin and Ito (as stochastic counterparts of deterministic chain rule) turn out to be the crucial tools to carry out a rigorous mathematical analysis. In fact, they provide existence, uniqueness, asymptotic properties and the link between continuous and discrete time stochastic dynamical systems. As a generalization, we obtain stochastic Taylor expansions of functionals of SDEs (Wagner-Platen formula). As in deterministic analysis, the latter formula is essential for the systematic construction of stochastic-numerical methods and an investigation of the qualitative behavior of their continuous time limits by approximating simulation.

In particular, if time permits, we present our own results in view of the general framework. So we arrive at the main principles of stochastic approximations (see also a second, more technical talk at the department), estimates of nonlinear contractivity and stability exponents, the class of balanced implicit methods, preservation of boundary conditions, asymptotically exact numerical methods, stochastic dissipativity, a fairly general stability analysis of stochastic systems and some convergence results.

We shall end with some applications in Engineering (reliability of stochastic systems, Lyapunov exponents, exit times), Mathematical Finance (asset and option pricing with random volatility) and Marketing Sciences (innovation diffusion). We are aiming at giving some overview rather than a too technical lecture and at making this lecture as simple as possible such that the interested audience can grasp some ideas about the possible range of further application fields up to the analysis of (random) time series as met in practically oriented data analysis.