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Teoriya Veroyatnostei i ee Primeneniya, 1981, Volume 26, Issue 3, Pages 597–606
(Mi tvp2607)
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This article is cited in 1 scientific paper (total in 1 paper)
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
Citation:
S. I. Pisanec, “Limit theorems for the processes of diffusion in $R^m$”, Teor. Veroyatnost. i Primenen., 26:3 (1981), 597–606; Theory Probab. Appl., 26:3 (1982), 587–594
Linking options:
https://www.mathnet.ru/eng/tvp2607 https://www.mathnet.ru/eng/tvp/v26/i3/p597
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