On the asymptotic behavior of the solution of a system of stochastic differential equations without aftereffect. (Russian) Zbl 0583.60056

Tonkikh, L. S.

Teor. Veroyatn. Mat. Stat. 31, 133-141 (1984).

Let \(\xi\) be an m-dimensional process satisfying a stochastic differential equation of the form \[ d\xi (t)=a(t,\xi (t))dt+\sigma (t,\xi (t))dw(t)+\int \gamma (t,\xi (t)){\tilde \nu}(dt,du),\quad t>0,\quad \xi (0)=\xi_ 0, \] where w is an m-dimensional Wiener process and \({\tilde \nu}\)(t,\(\cdot)\) is a centered Poisson measure, \(t\geq 0\). It is shown that under some conditions the sequence of processes \(\{\) \(\xi\) (nt)/\(\sqrt{n},t\geq 0\}\), \(n\geq 1\), is weakly convergent on [0,T] for every \(T>0\).

Reviewer: A.Ya.Dorogovtsev

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65	Brownian motion
60B10	Convergence of probability measures

Keywords:

weak convergence; Poisson measure