Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Optimal discretization of stochastic integrals driven by general Brownian semimartingale

Emmanuel Gobet and Uladzislau Stazhynski

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Abstract

We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretization stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable.

Résumé

Nous étudions l’erreur de discrétisation optimale d’intégrale stochastique, dirigée par une semimartingale brownienne continue multidimensionnelle. Dans ce cadre, nous déterminons une borne inférieure trajectorielle pour la variation quadratique de l’erreur renormalisée et nous fournissons une suite de temps d’arrêt de discrétisation, suite qui est asymptotiquement optimale. Cette dernière est définie explicitement à partir des temps d’atteinte d’ellipsoïdes aléatoires par la semimartingale sous-jacente. En comparaison avec les précédents résultats, nous considérons une très grande classe de semimartingales (relâchant en particulier les conditions de non dégénérescence qui étaient habituellement requises) et nous prouvons que la borne inférieure asymptotique est atteignable.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1556-1582.

Dates
Received: 2 January 2016
Revised: 20 March 2017
Accepted: 12 June 2017
First available in Project Euclid: 11 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1531296029

Digital Object Identifier
doi:10.1214/17-AIHP848

Mathematical Reviews number (MathSciNet)
MR3825891

Zentralblatt MATH identifier
06976085

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60F15: Strong theorems 60H05: Stochastic integrals

Keywords
Discretization of stochastic integrals Hitting times Random ellipsoids Almost sure convergence

Citation

Gobet, Emmanuel; Stazhynski, Uladzislau. Optimal discretization of stochastic integrals driven by general Brownian semimartingale. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1556--1582. doi:10.1214/17-AIHP848. https://projecteuclid.org/euclid.aihp/1531296029


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