## General theorems for numerical approximation of stochastic processes on the Hilbert space H2([0,T ],μ,Rd)

Henri Schurz

### Abstract

General theorems for the numerical approximation on the separable Hilbert space $H_2([0,T],\mu,{\bf R}^d)$ of cadlag, $({\cal F}_t)$-adapted stochastic processes with $\mu$-integrable second moments is presented for nonrandom intervals $[0,T]$ and positive measure $\mu$. The use of the theorems is illustrated by the special case of systems of ordinary stochastic differential equations (SDEs) and their numerical approximation given by the drift-implicit Euler method under one-sided Lipschitz-type conditions.

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### Key words

stochastic-numerical approximation, stochastic Lax-Theorem, ordinary stochastic differential equations, numerical methods, drift-implicit Euler methods, balanced implicit methods.

### AMS subject classifications

65C20, 65C30, 65C50, 60H10, 37H10, 34F05.

### Links to the cited ETNA articles

 [3] Vol. 11 (2000), pp. 131-151 Christopher T. H. Baker and Evelyn Buckwar: Continuous Θ-methods for the stochastic pantograph equation