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
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 795-802
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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. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 4, pp. 795-802. http://geodesic.mathdoc.fr/item/TVP_1982_27_4_a16/
@article{TVP_1982_27_4_a16,
author = {G. L. Kulini\v{c}},
title = {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},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {795--802},
year = {1982},
volume = {27},
number = {4},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_4_a16/}
}
TY - JOUR
AU - G. L. Kulinič
TI - 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
JO - Teoriâ veroâtnostej i ee primeneniâ
PY - 1982
SP - 795
EP - 802
VL - 27
IS - 4
UR - http://geodesic.mathdoc.fr/item/TVP_1982_27_4_a16/
LA - ru
ID - TVP_1982_27_4_a16
ER -
%0 Journal Article
%A G. L. Kulinič
%T 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
%J Teoriâ veroâtnostej i ee primeneniâ
%D 1982
%P 795-802
%V 27
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_1982_27_4_a16/
%G ru
%F TVP_1982_27_4_a16
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.
