Article Dans Une Revue Electronic Journal of Probability Année : 2024

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.

Dates et versions

hal-01323288 , version 1 (30-05-2016)

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Alexandre Richard, Denis Talay. Lipschitz continuity in the Hurst parameter of functionals of stochastic differential equations driven by a fractional Brownian motion. Electronic Journal of Probability, 2024, 29 (none), ⟨10.1214/24-EJP1191⟩. ⟨hal-01323288⟩
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