Stochastic differential equations of second order with a small parameter

TL;DR

This paper investigates second-order stochastic differential equations with a small parameter, establishing existence, uniqueness, and asymptotic behavior of solutions, including an averaging theorem and explicit limit forms as the parameter approaches zero.

Contribution

It provides new existence and uniqueness results and derives explicit asymptotic limits for solutions of second-order stochastic differential equations with small parameters.

Findings

Proved a special existence and unicity theorem for strong solutions.

Studied the asymptotic behavior of solutions as the small parameter tends to zero.

Established the stochastic averaging theorem and explicit limit forms for solutions.

Abstract

We consider boundary value problems for stochastic differential equations of second order with a small parameter. For this case we prove a special existence and unicity theorem for strong solutions. The asymptotic behavior of these solutions as small parameter goes to zero is studied. The stochastic averaging theorem for such equations is shown. The limits in the explicit form for the solutions as a small parameter goes to zero are found.