Non-degeneracy of Wiener functionals arising from rough differential equations

TL;DR

This paper combines Malliavin Calculus and rough path analysis to establish the existence of densities for solutions to stochastic differential equations driven by a broad class of non-degenerate Gaussian processes, even with irregular sample paths.

Contribution

It introduces a novel approach merging Malliavin Calculus and rough path theory to analyze the regularity and density of solutions driven by complex Gaussian processes.

Findings

Existence of densities for solutions driven by non-degenerate Gaussian processes

Extension of regularity results to processes with irregular sample paths

Unified framework for analyzing SDEs with rough signals

Abstract

Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Ito map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss existence of a density for solutions to stochastic differential equations driven by a general class of non-degenerate Gaussian processes, including processes with sample path regularity worse than Brownian motion.