Stochastic symmetries and transformations of stochastic differential equations

TL;DR

This paper introduces the concept of stochastic symmetries for differential equations, providing conditions for their existence and methods to transform complex stochastic equations into solvable forms.

Contribution

It defines stochastic symmetries, derives necessary conditions involving infinitesimal generators, and offers a framework for transforming stochastic differential equations into solvable target equations.

Findings

Defined stochastic symmetries for differential equations

Derived necessary conditions involving infinitesimal generators

Presented methods to transform equations into solvable forms

Abstract

In this article, we introduce the notion of stochastic symmetry of a differential equation. It consists in a stochastic flow that acts over a solution of a differential equation and produces another solution of the same equation. In the ordinary case, we give necessary conditions in order to obtain such symmetries. These conditions involve the infinitesimal generator of the flow and the coefficients of the equation. Moreover, we show how to obtain necessary conditions in order to find an application that transforms a stochastic differential equation that one would like to solve into a target equation that one previously know how to solve.