A limit theorem for stochastic difference schemes with random coefficients.(English. Russian original)Zbl 0635.60069

Theory Probab. Math. Stat. 33, 1-9 (1986); translation from Teor. Veroyatn. Mat. Stat. 33, 3-11 (1985).
Consider the following stochastic difference scheme: $\Delta \xi_{ni}=a_{ni}(\xi_{ni},z_{ni})\Delta t_{ni}+b_{ni}(\xi_{ni},z_{ni})\Delta \psi_{ni},\quad i=0,...,m_ n,\quad n=1,2,...$ where $$\Delta u_{ni}=u_{ni+1}- u_{ni}$$ $$(u=t,\xi,\psi)$$, $$0=t_{n_ 0}<t_{n_ 1}...<t_{nm_ n+1}=T$$, the random variables $$z_{ni}\in {\mathcal F}_{ni}\uparrow$$, and $$(\psi_{ni},{\mathcal F}_{ni})$$ is a martingale. Put $$\xi_ n(t)=\xi_{n0}+\sum_{t_{ni}+1\leq t}\Delta \xi_{ni}.$$
It is shown, under some regularity conditions on $$\xi_ n(.)$$ (sublinearity and Lipschitzianity of A(t,x),B(t,x)) that $$\xi_ n(.)$$ converges weakly in D[0,T] to the solution of the Ito stochastic differential equation $d\xi (t)=A(t,\xi (t))dt+B(t,\xi (t))dW(t),\quad \xi (0)=\xi_ 0,$ where W(t) is a Brownian motion.
Reviewer: A.Korzeniowski

MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 39A10 Additive difference equations 60F05 Central limit and other weak theorems