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

## TL;DR

This paper investigates how the probability distributions of functionals of solutions to fractional Brownian motion driven stochastic differential equations change as the Hurst parameter approaches 1/2, extending Gaussian density estimates uniformly.

## Contribution

It introduces a novel sensitivity analysis method for SDEs driven by fractional Brownian motion, focusing on the limit as H approaches 1/2, with uniform Gaussian estimates.

## Key findings

- Extended Gaussian density estimates uniformly in time and Hurst parameter.
- Analyzed the sensitivity of first passage times as H approaches 1/2.
- Provided new perspectives on fractional Brownian motion driven SDEs.

## 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 $\{X^H_t\}_{t\in \mathbb{R}_+}$ 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 $X^H_t$ to estimates with constants which are uniform w.r.t. $t$ in in the whole half-line $\R_+-\{0\}$ and $H$ when $H$ tends to~$\tfrac{1}{2}$.