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

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-t(H)}(t is an element of 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 = 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, the 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 1/2. Moreover, we use our accurate estimates on Laplace transforms to get a weak convergence rate of the first passage times of X-H when H tends to 1/2. Our results show that the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to 1/2.