Integration by parts formulas and Lie's symmetries of SDEs

TL;DR

This paper develops integration by parts formulas for stochastic differential equations using Lie's symmetry theory, connecting stochastic calculus, geometry, and analysis, with applications to Brownian motion models.

Contribution

It introduces a novel approach applying Lie's symmetries to derive integration by parts formulas for SDEs, extending Malliavin calculus techniques.

Findings

Established a strong quasi-invariance principle for SDEs.

Derived finite-dimensional integration by parts formulas.

Applied the theory to specific Brownian motion driven models.

Abstract

A strong quasi-invariance principle and a finite-dimensional integration by parts formula as in the Bismut approach to Malliavin calculus are obtained through a suitable application of Lie's symmetry theory to autonomous stochastic differential equations. The main stochastic, geometrical and analytical aspects of the theory are discussed and applications to some Brownian motion driven stochastic models are provided.