Hölder continuity in the Hurst parameter of functionals of Stochastic Differential Equations driven by fractional Brownian motion

Alexandre Richard, Denis Talay

Published 2016-05-11Version 1

In this paper we develop sensitivity analyses w.r.t. the long-range/memory noise parameter for solutions to stochastic differential equations and the probability distributions of their first passage times at given thresholds. Here we consider the case of stochastic differential equations driven by fractional Brownian motions and the sensitivity, when the Hurst parameter~$H$ of the noise tends to the pure Brownian value, of probability distributions of certain functionals of the trajectories of the solutions $\{X^H_t\}_{t\in \R_+}$. We first get accurate sensitivity estimates w.r.t. $H$ around the critical Brownian parameter $H=\tfrac{1}{2}$ of time marginal probability distributions of $X^H$. We second develop a sensitivity analysis for the Laplace transform of first passage time of $X^H$ at a given threshold. Our technique requires accurate Gaussian estimates on the density of $X^H_t$. The Gaussian estimate we obtain in Section~5 may be of interest by itself.