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

S. I. Pisanec

Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 3, pp. 597-606

Résumé

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 $$ \alpha\eta_t=\beta_t(\eta)\,dw_t,\qquad \eta_0=\xi_0,\qquad t\le T. $$