Smoothness of densities for path-dependent SDEs under H\"ormander's condition

TL;DR

This paper proves the existence of smooth probability densities for solutions to a class of path-dependent SDEs under H"ormander's condition, using a novel lifting approach and advanced calculus techniques.

Contribution

It introduces a new method to establish density smoothness for path-dependent SDEs by lifting into Banach spaces and applying a H"ormander-type bracket condition.

Findings

Existence of smooth densities under H"ormander's condition for path-dependent SDEs.

A novel lifting technique into Banach spaces for path-dependent SDE analysis.

Integration of Malliavin calculus and rough path theory in the proof.

Abstract

We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a H\"ormander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given $n$-dimensional path-dependent SDE into a suitable $L_p$-type Banach space in such a way that the lifted Banach-space-valued equation becomes a state-dependent reformulation of the original SDE. We then formulate H\"ormander's bracket condition in $\mathbb R^n$ for non-anticipative SDE coefficients defining the Lie brackets in terms of vertical derivatives in the sense of the functional It\^o calculus. Our pathway to the main result engages an interplay between the analysis of SDEs in Banach spaces, Malliavin calculus, and rough path techniques.