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Short Communications

Limit theorems for the processes of diffusion in $R^m$

S. I. Pisanec

Kiev

Abstract: Let $\xi_t^{(n)}$ ($n=0,1,\dots$) be a sequence of solutions of stochastic differential equations
$$ d\xi_t^{(n)}=\alpha_t^{(n)}(\xi^{(n)})dt+\beta_t(\xi^{(n)})dw_t,\qquad \xi_0^{(n)}=\xi_0,\ 0\le t\le T,\ n=0,1,\dots $$
In the paper we study the conditions which are sufficient for
$$ \lim_{n\to\infty}\mathbf M|\xi_t^{(n)}-\xi_t^{(0)}|^2=0,\qquad t\le T, $$
or for
$$ \lim_{n\to\infty}\mathbf M\biggl|\int_0^t\alpha_s^{(n)}(\eta)\,ds- \int_0^t\alpha_s^{(0)}(\eta)\,ds\biggr|^2=0,\qquad t\le T, $$
where $\eta_t$ is the solution of an equation
$$ \alpha\eta_t=\beta_t(\eta)\,dw_t,\qquad \eta_0=\xi_0,\qquad t\le T. $$


Received: 31.07.1978

 English version: Theory of Probability and its Applications, 1982, 26:3, 587–594

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