Symmetry and integrability for stochastic differential equations

TL;DR

This paper explores the relationship between symmetries and integrability of stochastic differential equations, extending previous results and considering both deterministic and random symmetries to understand reducibility and solution methods.

Contribution

It extends existing theories by analyzing symmetry-integrability relations for SDEs, including new insights into reducibility and the role of random symmetries.

Findings

Symmetries are linked to the integrability of SDEs.

Both deterministic and random symmetries influence reducibility.

The paper generalizes previous results by Kozlov and others.

Abstract

We discuss the interrelations between symmetry of an Ito stochastic differential equations (or systems thereof) and its integrability, extending in party results by R. Kozlov [J. Phys. A ${\bf 43}$ (2010) \& ${\bf 44}$ (2011)]. Together with integrability, we also consider the relations between symmetries and reducibility of a system of SDEs to a lower dimensional one. We consider both "deterministic" symmetries and "random" ones, in the sense introduced recently by Gaeta and Spadaro [J.Math. Phys. ${\bf 58}$ (2017)].