Noise Sensitivity of Functionals of Fractional Brownian Motion Driven Stochastic Differential Equations: Results and Perspectives

Abstract

We present an innovating sensitivity analysis for stochastic differential equations: We study the sensitivity, when the Hurst parameter H of the driving fractional Brownian motion tends to the pure Brownian value, of probability distributions of smooth functionals of the trajectories of the solutions {XtH}t∈R+ and of the Laplace transform of the first passage time of XH at a given threshold. Our technique requires to extend already known Gaussian estimates on the density of XtH to estimates with constants which are uniform w.r.t. t in the whole half-line R+- { 0 } and H when H tends to 12.