A Study of a Class of Stochastic Volterra Equations Driven by Fractional Brownian Motion

TL;DR

This paper investigates stochastic Volterra equations driven by fractional Brownian motion, establishing integration by parts, Harnack inequalities, and transportation cost inequalities to analyze solution regularities and properties.

Contribution

It introduces new analytical tools like the Driver type formula and Bismut derivative formula for these equations, advancing understanding of their solution laws.

Findings

Established integration by parts and Harnack inequalities.

Derived Bismut type derivative formula using Malliavin calculus.

Proved transportation cost inequalities for the solution law.

Abstract

This paper is devoted to study a class of stochastic Volterra equations associated with fractional Brownian motion. We first prove the Driver type integration by parts formula and the shift Harnack type inequalities. As a direct application, we provide an alternative method to describe the regularities of the law of the solution. Secondly, by using the Malliavin calculus, the Bismut type derivative formula is established, which is then applied to the study of the gradient estimate and the strong Feller property. Finally, we establish the Talagrand type transportation cost inequalities for the law of the solution on the path space with respect to both the uniform metric and the $L^2$-metric.