Existence and smoothness of the density of the solution to fractional stochastic integral Volterra equations

TL;DR

This paper proves the existence and smoothness of the probability density for solutions to fractional stochastic Volterra equations driven by fractional Brownian motion with Hurst parameter greater than 1/2, extending previous results.

Contribution

It extends prior work on stochastic differential equations to more general Volterra equations driven by fractional Brownian motion, providing new regularity results.

Findings

Derived supremum norm estimates for solutions and their Malliavin derivatives.

Proved existence and smoothness of the solution's density under nondegeneracy conditions.

Extended previous results to a broader class of stochastic Volterra equations.

Abstract

We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 . We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and smoothness of the density under suitable nondegeneracy conditions. This extends the results in Hu-Nualart and Nualart-Saussereau where stochastic differential equations driven by fractional Brownian motion are considered. The proof uses a priori estimates for deterministic differential equations driven by a function in a suitable Sobolev space.