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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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