Malliavin calculus and densities for chaos-driven stochastic differential equations

TL;DR

This paper develops a Malliavin calculus framework for chaos-driven stochastic differential equations with non-Gaussian noise, establishing solution existence, uniqueness, and density properties.

Contribution

It introduces a novel approach using the Kusuoka-Stroock method for Malliavin calculus on chaos processes, extending analysis beyond Gaussian settings.

Findings

Established existence and uniqueness of solutions under mild conditions.

Proved Malliavin differentiability and absolute continuity of solutions.

Derived density results for solutions under ellipticity and non-degeneracy assumptions.

Abstract

We study stochastic differential equations driven by finite-order chaos processes on abstract Wiener spaces, with pathwise Riemann-Stieltjes integration. The driving noise is an $\mathbb{R}^m$-valued chaotic process given by multiple Wiener-It\^o integrals of fixed order, allowing for non-Gaussian dynamics. Under mild smoothness assumptions on the coefficients and H\"older-type regularity of the noise, we establish existence and uniqueness of solutions. We then prove Malliavin differentiability and absolute continuity of the law of the solution. Since the usual Gaussian isonormal framework is unavailable, we rely on the Kusuoka-Stroock approach to Malliavin calculus and develop a Taylor expansion for multiple integrals under Cameron-Martin shifts. Under suitable ellipticity, independence, and non-degeneracy conditions, the Bouleau-Hirsch criterion yields density results. Applications to multidimensional Hermite-driven equations are provided.