An axiomatic approach to numerical approximations of stochastic processes

An axiomatic approach to the numerical approximation Y of some stochastic process X with values on a separable Hilbert space H is presented by means of Lyapunov-type control functions V. The processes X and Y are interpreted as flows of stochastic differential and difference equations, respectively. The main result is the proof of some extensions of well-known deterministic principle of Kantorovich-Lax-Richtmeyer to approximate solutions of initial value differential 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 drift-implicit backward Enter methods applied to ordinary stochastic differential equations (SDEs) driven by standard Wiener processes on Euclidean spaces H = R-d along functions such as V(x) = Sigma(k)(i=0) c(i)x(2i). A detailed discussion on an example with cubic nonlinearity from field theory in physics (stochastic Ginzburg-Landau equation) illustrates the suggested axiomatic approach.