Reduction and reconstruction of stochastic differential equations via symmetries

TL;DR

This paper introduces an algorithmic approach to reduce and reconstruct stochastic differential equations using symmetries, providing a new perspective and solutions for linear cases, with applications demonstrated through examples.

Contribution

It presents a novel symmetry-based method for reducing and reconstructing stochastic differential equations, including a new solution formula for linear cases.

Findings

Effective symmetry-based reduction algorithm developed

Reconstruction method inspired by classical quadratures introduced

Solution formula for linear SDEs derived within this framework

Abstract

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance.