Integrable Ito equations with multiple noises

TL;DR

This paper classifies and integrates scalar Ito equations with multiple independent noises that admit standard symmetries, extending previous single-noise results to two noises and showing no symmetries with three or more noises.

Contribution

It extends the classification and integration of scalar Ito equations with standard symmetries from one noise to two noises, and demonstrates the absence of such symmetries with three or more noises.

Findings

All two-noise scalar Ito equations with standard symmetries are classified.

Such equations can be integrated using the Kozlov construction.

No standard symmetries exist for equations with three or more noises.

Abstract

The classification of scalar Ito equations with a single noise source which admit a so called standard symmetry and hence are -- by the Kozlov construction -- integrable is by now complete. In this paper we study the situation, occurring in physical as well as biological applications, where there are two independent noise sources. We determine all such autonomous Ito equations admitting a standard symmetry; we then integrate them by means of the Kozlov construction. We also consider the case of three or more independent noises, showing no standard symmetry is present.