MATHEMATICS COLLOQUIUM

 WHEN:
On Friday, February 04, 2000 at 2:00 PM

WHERE:
Durham Science Center, Room 255

WHAT:

Henri Schurz
School of Mathematics
University of Minnesota, Minneapolis

will give a talk on

The main theorem of numerical approximation theory for stochastic processes on randomized Banach spaces

Abstract:
A fairly general concept of main principles to approximate stochastic processes (fields) on appropriately randomized Banach spaces shall be presented. We introduce the concepts of D--invariance (a.s.), mean and p-th mean consistency, numerical p-th mean stability, contractivity and dissipativity of stochastic processes with values in one and the same Banach space. In a unique way we combine all these principles to show the p-th mean convergence of two F_t--adapted processes (X,Y) satisfying these natural requirements. We also give a stochastic variant of well-known deterministic Lax-Richtmeyer equivalence theorem which locally holds in stochastic settings too. A worst case convergence rate is also estimated in terms of local mean and $p$-th mean consistency rates. The presented analysis culminates in an adequateness diagramme for (stochastic) approximation theory. Eventually, we apply the general ideas to the example of moment-dissipative, nonlinear, stochastic differential equations (SDEs) with polynomial drift and diffusion parts and their numerical analysis on any finite time-interval, illustrated with some easily understandable test SDEs from a rich potential range of applications coming from Mathematical Biology, Physics, Chemistry, Engineering, Finance and Marketing. Moreover, we will provide very efficient approximation error estimates which allow to form nontrivial statements on the accuracy of approximations on infinite time-intervals and some sharp estimates for some autonomous, linear SDEs.