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On the approximation of stochastic differential equations. (English) Zbl 0635.60071
This paper contains results on the stability properties of the stochastic differential equation $X(t)=\xi +\int_{(0,t]}\sigma (s,\omega,X(s))\cdot dM(s)$ with respect to perturbations of the continuous multidimensional semi-martingale M and the matrix-valued random field ($$\sigma$$ (s,$$\omega$$,x)) in the topology of uniform convergence in probability. The stochastic integral is understood in the Stratonovich sense.
Previous results are generalized by allowing $$\sigma$$ to be random and to be not necessarily bounded with continuous not necessarily bounded derivatives up to the second order only. Results are presented also for the stochastic differential equation $Z(t)=\xi +\int_{(0,t]}\sigma (s,\omega,Z(s))*dM(s),$ where the stochastic integral is understood in the sense $\int_{(0,t]}\sigma (s,\omega,Z(s))*dM(s):=\lim_{n\to \infty}\sum_{t^ n_{i+1}\leq t}2^{-1}(\sigma (t^ n_{i+1},Z(t^ n_ i))+\sigma (t^ n_{i+1},Z(t^ n_{i+1}))(M(t^ n_{i+1})- M(t^ n_ i)),$ provided the limit exists in probability for every sequence of partitions $$\{0=t^ n_ 0\leq t^ n_ 1\leq...\leq t^ n_ N=t\}$$ such that $$\lim_{n\to \infty}\max_{i}| t^ n_{i+1}-t^ n_ i| =0$$.
Reviewer: L.G.Gorostiza

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