Lipschitz continuity in the Hurst parameter of functionals of stochastic differential equations driven by a fractional Brownian motion
Résumé
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 {XHt}t∈\R+ to stochastic differential equations driven
by frac{t}ional Brownian motions.
We develop two innovative sensitivity analyses
when the Hurst parameter~H of the noise tends to the
critical Brownian parameter H=12 from above or from
below. First, we examine expected smooth functions of XH 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 XH at a given threshold.
In both cases we exhibit the Lipschitz continuity w.r.t.~H around the
value 12. Therefore, our results show that the
Markov Brownian model is a good proxy
model as long as the Hurst parameter remains close to~12.