Random rotational invariance of integration by parts formulas within a Bismut-type approach

TL;DR

This paper proves the rotational invariance of an integration by parts formula inspired by Bismut's approach within the Lie symmetry framework, highlighting the effect of Brownian motion invariance in stochastic models.

Contribution

It introduces a stochastic rotational invariance property of an integration by parts formula using Lie symmetry theory, extending Bismut's approach.

Findings

Proves rotational invariance of the integration by parts formula

Demonstrates invariance in explicit 2D Brownian models

Discusses the effect of Brownian motion invariance

Abstract

The stochastic rotational invariance of an integration by parts formula inspired by the Bismut approach to Malliavin calculus is proved in the framework of the Lie symmetry theory of stochastic differential equations. The non-trivial effect of the rotational invariance of the driving Brownian motion in the derivation of the integration by parts formula is discussed and the invariance property of the formula is shown via applications to some explicit two-dimensional Brownian motion-driven stochastic models.