Smoothness and Gaussian Density Estimates for Stochastic Functional Differential Equations with Fractional Noise

Nguyen Van Tan

doi:10.19139/soic-2310-5070-784

Abstract

In this paper, we study the density of the solution to a class of stochastic functional differential equations driven by fractional Brownian motion. Based on the techniques of Malliavin calculus, we prove the smoothness and establish upper and lower Gaussian estimates for the density.

Highlights

In the last decade, Gaussian density estimates for the solutions of various stochastic equations have been intensively studied
The class of stochastic equations with fractional noise has been discussed by several authors, see [1, 2, 4, 8] and references therein
We recall that fractional Brownian motion of Hurst parameter H ∈ (0, 1) is a centered Gaussian process BH = (BtH )t∈R+ with covariance function

Summary

Introduction

Gaussian density estimates for the solutions of various stochastic equations have been intensively studied. We consider stochastic functional differential equations of the form [. The density of solutions to the equation has been discussed in some special cases. BH reduces to standard Brownian motion and in this case, the existence and smoothness of the probability density of solutions were proved by Takeuchi in [11]. We would like to emphasize that the complexity of stochastic integrals with respect to fBm and the appearance of delayed integral term in (2) require a fine analysis for proving the properties (i) and (ii).

Preliminaries

The main results

Findings

Conclusion