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

Abstract

We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter . 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 and Nualart [Differential equations driven by Hölder continuous functions of order greater than 1/2, Stoch. Anal. Appl. Abel Symp. 2 (2007), pp. 399–413] and Nualart and Saussereau [Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stoch. Process. Appl. 119 (2009), pp. 391–409] 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.