Positivity of the density for rough differential equations

TL;DR

This paper investigates the conditions under which the probability density of solutions to rough differential equations is strictly positive, building on recent advances in Malliavin calculus for such equations.

Contribution

It applies the Aida-Kusuoka-Stroock theory to establish criteria for the positivity of densities in rough differential equations, extending previous smoothness results.

Findings

Established conditions for strict positivity of densities

Connected Malliavin calculus with positivity criteria

Extended theoretical framework for rough differential equations

Abstract

Due to recent developments of Malliavin calculus for rough differential equations, it is now known that, under natural assumptions, the law of a unique solution at a fixed time has a smooth density function. Therefore, it is quite natural to ask whether or when the density is strictly positive. In this paper we study this problem from the viewpoint of Aida-Kusuoka-Stroock's general theory.