Noise sensitivity of functionals of fractional Brownian motion driven stochastic differential equations: Results and perspectives

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 $\{X_t^H{\}}_{t\in {\mathbb{R}}_{+}}$ and of the Laplace transform of the first passage time of $X^H$ at a given threshold. Our technique requires to extend already known Gaussian estimates on the density of $X_t^H$ to estimates with constants which are uniform w.r.t. $t$ in in the whole half-line ${\mathbb{R}}_{+}-\{0\}$ and $H$ when $H$ tends to~$\frac{1}{2}$.