Short Communications

On the strong convergence of the sequence of diffusion type processes

S. I. Pisanec

Institute of Cybernetics AS Ukrainian SSR, Kiev

Abstract: Let $\xi_t^{(n)}$, $t\le T$ be the sequence of solutions of stochastic differential equations
$$ d\xi_t^{(n)}=\alpha_t^{(n)}(\xi_t^{(n)})\,dt+dw_t,\qquad\xi_t^{(n)}=0,\qquad n=0,1,\dots $$
In this paper we study the conditions under which
$$ \lim_{n\to\infty}\mathbf M\biggl|\int_0^t\alpha_s^{(n)}(w)\,ds- \int_0^t\alpha_s^{(0)}(w)\,ds\biggr|^2=0,\qquad t\le T, $$
and the conditions under which
$$ \lim_{n\to\infty}\mathbf M|\xi_t^{(n)}-\xi_t^{(0)}|^2=0,\qquad t\le T. $$


Received: 10.10.1977

 English version: Theory of Probability and its Applications, 1980, 24:4, 863–869

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