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

Anisimov, V. V.; Yurachkovskij, A. P.

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

Keywords:

weak convergence; stochastic difference scheme; Ito stochastic differential equation; Brownian motion