Symmetries of Kolmogorov backward equation

TL;DR

This paper explores the relationship between symmetries and first integrals of stochastic differential equations and their associated Kolmogorov equations, highlighting connections between forward and backward symmetries.

Contribution

It establishes new links between symmetries of Itô SDEs and their Kolmogorov equations, including forward and backward forms.

Findings

Identifies the relation between symmetries and first integrals of Itô SDEs.

Connects symmetries of Kolmogorov backward and forward equations.

Provides a framework for analyzing symmetries in stochastic differential equations.

Abstract

The note provides the relation between symmetries and first integrals of It\^{o} stochastic differential equations and symmetries of the associated Kolmogorov backward equation. Relation between the symmetries of the Kolmogorov backward equation and the symmetries of the Kolmogorov forward equation is also given.