Limit theorems for the processes of diffusion in $R^m$
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 3, pp. 597-606
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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
$$
\alpha\eta_t=\beta_t(\eta)\,dw_t,\qquad \eta_0=\xi_0,\qquad t\le T.
$$
@article{TVP_1981_26_3_a12,
author = {S. I. Pisanec},
title = {Limit theorems for the processes of diffusion in $R^m$},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {597--606},
publisher = {mathdoc},
volume = {26},
number = {3},
year = {1981},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_3_a12/}
}
S. I. Pisanec. Limit theorems for the processes of diffusion in $R^m$. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 3, pp. 597-606. http://geodesic.mathdoc.fr/item/TVP_1981_26_3_a12/