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A note on approximation for stochastic differential equations. (English) Zbl 0647.60072
Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 155-162 (1988).
[For the entire collection see Zbl 0635.00013.]
Let \[ (1)\quad dX(t)=\sigma (t,X(t))dB(t)+b(t,X(t))dt \] be a stochastic differential equation on \(R^ d\) with pathwise unique solution \(X(t,x)\) for each initial value \(x\in R^ d\). We consider a sequence of stochastic differential equations \[ (2)\quad dX_ n(t)=\sigma_ n(t,X_ n(t))dB(t)+b_ n(t,X_ n(t))dt \] with continuous coefficients which enjoy \[ \sup_{n}\sup_{t\leq T}(\| \sigma_ n(t,x)\| +| b_ n(t,x)|)\leq L_ T(1+| x|)\text{ for each } T\geq 0, \] where \(L_ T\) denotes a constant depending on T. In this setting, we assume that each equation of (2) has a solution \(X_ n(t,x)\) for any starting point \(x\in R^ d\). We prove that if (2) is an approximating sequence, for (1) in the sense \[ \lim_{n\to \infty}\sup_{t\leq T}\sup_{x\in K}(\| \sigma_ n(t,x)-\sigma (t,x)\| +| \quad b_ n(t,x)-b(t,x)|)=0 \] for each \(T\geq 0\) and compact set K in \(R^ d\) (hence \(\sigma(t,x)\) and \(b(t,x)\) become continuous), then \[ \lim_{n\to \infty}\sup_{x\in K}E\left[\max_{0\leq t\leq T}| X_ n(t,x)-X(t,x)|^ 2\right]=0 \] holds for each \(T\geq 0\) and each compact set K in \(R^ d\).
Reviewer: S.Nakao

MSC:
60H20 Stochastic integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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