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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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