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Teoriya Veroyatnostei i ee Primeneniya, 1982, Volume 27, Issue 4, Pages 795–802
(Mi tvp2437)
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This article is cited in 8 scientific papers (total in 8 papers)
Short Communications
On necessary and sufficient conditions for the convergence of solutions of one-dimensional diffusion stochastic equations with a non-regular dependence of coefficients on a parameter
G. L. Kulinič Kiev
Abstract:
We consider an one-dimensional stochastic differential equation of diffusion type
$$
d\xi_\alpha(t)=a_\alpha(\xi_\alpha(t))\,dt+\sigma_\alpha(\xi_\alpha(t))\,dw_\alpha(t),\qquad t>0.
$$
where $\alpha>0$ is a parameter, $a_\alpha(x)$, $\sigma_\alpha(x)>0$ are real functions which may degenerate at some points $x_k$ as $\alpha\to 0$ and $w_\alpha(t)$ is a family of Wiener processes. The necessary and sufficient conditions for the weak convergence of $\xi_\alpha(t)$ to the generalized diffusion process $\alpha\to 0$ are obtained.
Received: 01.04.1980
Citation:
G. L. Kulinič, “On necessary and sufficient conditions for the convergence of solutions of one-dimensional diffusion stochastic equations with a non-regular dependence of coefficients on a parameter”, Teor. Veroyatnost. i Primenen., 27:4 (1982), 795–802; Theory Probab. Appl., 27:4 (1983), 856–862
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