A Milstein-type scheme without Lévy area terms for SDEs driven by fractional brownian motion

Deya, A.; Neuenkirch, A.; Tindel, S.

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 518-550.

Nous étudions dans cet article l'approximation numérique d'équations différentielles dirigées par un mouvement brownien fractionnaire (mBf) de coefficient de Hurst H > 1/3. L'algorithme effectif que nous proposons repose sur un développement au second ordre, où l'aire de Lévy est remplacée par un produit d'incréments du mBf. Nous obtenons la convergence de notre schéma en combinant des méthodes issues de la théorie des trajectoires rugueuses et des résultats sur l'approximation de l'aire de Lévy.

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second-order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretization of the Lévy area terms.

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