Lipschitz continuity in the Hurst parameter of functionals of stochastic differential equations driven by a fractional Brownian motion

TL;DR

This paper investigates how the distributions of solutions to stochastic differential equations driven by fractional Brownian motion change as the Hurst parameter approaches 1/2, demonstrating Lipschitz continuity and the validity of Brownian models near this critical point.

Contribution

The paper introduces two sensitivity analysis methods for the Hurst parameter near 1/2, establishing Lipschitz continuity for functionals of solutions to fractional SDEs.

Findings

Lipschitz continuity of functionals w.r.t. H around 1/2

Validation of Brownian models near H=1/2

Applicability to irregular functionals like first passage times

Abstract

Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications. In this paper we consider solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$ to stochastic differential equations driven by fractional Brownian motions. We develop two innovative sensitivity analyses when the Hurst parameter $H$ of the noise tends to the critical Brownian parameter $H=\tfrac{1}{2}$ from above or from below. First, we examine expected smooth functions of $X^H$ at a fixed time horizon $T$. Second, we examine Laplace transforms of functionals which are irregular with regard to Malliavin calculus, namely, first passage times of $X^H$ at a given threshold. In both cases we exhibit the Lipschitz continuity w.r.t. $H$ around the value $\tfrac{1}{2}$. Therefore, our results show that the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to $\tfrac{1}{2}$.