Reduction and reconstruction of SDEs via Girsanov and quasi Doob symmetries

TL;DR

This paper introduces a reduction and reconstruction framework for stochastic differential equations using Girsanov and quasi Doob symmetries, providing new theoretical insights and practical applications.

Contribution

It proposes a novel reduction method for SDEs based on stochastic symmetries, including Girsanov transformations, and introduces a new reconstruction concept with a corresponding theorem.

Findings

Reconstruction involves expectation values of solution functionals.

A general reconstruction theorem for stochastic symmetries is established.

Applications to relevant stochastic models demonstrate the theoretical results.

Abstract

A reduction procedure for stochastic differential equations based on stochastic symmetries including Girsanov random transformations is proposed. In this setting, a new notion of reconstruction is given, involving the expectation values of functionals of solution to the SDE and a reconstruction theorem for general stochastic symmetries is proved. Moreover, the notable case of reduction under the closed subclass of quasi Doob transformations is presented. The theoretical results are applied to stochastic models relevant in the applications.