Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise

TL;DR

This paper classifies scalar autonomous Ito stochastic differential equations with simple noise based on their symmetries, extending previous work by including recent types of symmetries such as standard random symmetries, and provides integration methods for these equations.

Contribution

It offers a comprehensive classification of symmetric scalar Ito SDEs with simple noise, including new symmetry types like standard random symmetries, and provides their integration procedures.

Findings

Classification of symmetric equations with simple noise

Inclusion of standard random symmetries in the classification

Explicit integration methods for the classified equations

Abstract

It is known that knowledge of a symmetry of a scalar Ito stochastic differential equations leads, thanks to the Kozlov substitution, to its integration. In the present paper we provide a classification of scalar autonomous Ito stochastic differential equations with simple noise possessing symmetries; here "simple noise" means the noise coefficient is of the form $\s (x,t) = s x^k$, with $s$ and $k$ real constants. Such equations can be taken to a standard form via a well known transformation; for such standard forms we also provide the integration of the symmetric equations. Our work extends previous classifications in that it also consider recently introduced types of symmetries, in particular standard random symmetries, not considered in those.