On Lie-point symmetries for Ito stochastic differential equations

TL;DR

This paper investigates the behavior of Lie-point symmetries in Ito stochastic differential equations, establishing conditions under which these symmetries are preserved under coordinate transformations, especially for simple symmetries.

Contribution

It extends symmetry analysis to Ito SDEs by identifying when symmetries are coordinate-invariant, confirming Kozlov's theory on symmetry and integrability.

Findings

Simple symmetries are preserved under coordinate changes.

Symmetries relevant to Kozlov's theory are invariant under transformations.

The study clarifies the geometric nature of symmetries in stochastic equations.

Abstract

In the deterministic realm, both differential equations and symmetry generators are geometrical objects, and behave properly under changes of coordinates; actually this property is essential to make symmetry analysis independent of the choice of coordinates and applicable. When trying to extend symmetry analysis to stochastic (Ito) differential equations, we are faced with a problem inherent to their very nature: they are not geometrical object, and they behave in their own way (synthesized by the Ito formula) under changes of coordinates. Thus it is not obvious that symmetries are preserved under a change of coordinates. We will study when this is the case, and when it is not; the conclusion is that this is always the case for so called \emph{simple} symmetries. We will also note that Kozlov theory relating symmetry and integrability for stochastic differential equations is confirmed by our considerations and results, as symmetries of the type relevant in it are indeed of the type preserved under coordinate changes.