Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

A. Deya, A. Neuenkirch, and S. Tindel

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 2 (2012), 518-550.

Dates
First available in Project Euclid: 11 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1334148209

Digital Object Identifier
doi:10.1214/10-AIHP392

Mathematical Reviews number (MathSciNet)
MR2954265

Zentralblatt MATH identifier
1260.60135

Subjects
Primary: 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 65C30: Stochastic differential and integral equations

Keywords
Fractional Brownian motion Lévy area Approximation schemes

Citation

Deya, A.; Neuenkirch, A.; Tindel, S. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 2, 518--550. doi:10.1214/10-AIHP392. https://projecteuclid.org/euclid.aihp/1334148209


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