On weak convergence of stochastic differential equations with irregular coefficients

Abstract

One-dimensional Ito stochastic differential equations and stochastic differential equations with local time that depend on a small parameter ε and have irregular coefficients (for example, the unlimited drift coefficient or the non-Lipschitz diffusion coefficient) have been considered. The available results concerning the conditions for a mutually univocal correspondence between stochastic Ito equations and stochastic equations with local time were generalized. The weak convergence of the solutions of those equations at ε → 0 were analyzed. The form of the coefficients for the limiting process was obtained. The necessary and sufficient conditions for the weak convergence of the solutions of those equations to the limit random process were proved.