An Axiomatic Approach to Numerical Approximation of Stochastic Processes

Abstract

An axiomatic approach to the numerical approximation of stochastic processes X,Y with values on separable Hilbert spaces H is presented. The processes X and Y can be interpreted as flows of stochastic differential and difference equations, respectively. The main result of this talk deals with an extension of the well-known deterministic principles of Kantorovich-Lax-Richtmeyer to approximate solutions of differential initial value problems to the stochastic case. The concepts of invariance, smoothness of martingale parts, consistency, stability, and contractivity of stochastic processes are uniquely combined to derive efficient convergence rates on finite and infinite time-intervals. The applicability of our results is explained with the case of ordinary stochastic differential equations and drift-implicit Euler methods (i.e. backward Euler in drift term) driven by standard multi-dimensional Wiener processes on Euclidean spaces H = R^d and invariant subsets D.