Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Pathwise stochastic calculus with local times

Mark Davis, Jan Obłój, and Pietro Siorpaes

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Abstract

We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the usual (stochastic) local times a.s. for paths of a continuous semimartingale. We establish pathwise version of the Tanaka–Meyer, change of variables and change of time formulae. We provide equivalent conditions for existence of pathwise local time. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choice of partitions. In particular, we show that an arbitrary given non-decreasing process can be achieved a.s. by the pathwise quadratic variation of a standard Brownian motion for a suitable sequence of (random) partitions; however, such degenerate behaviour is excluded when the partitions are constructed from stopping times.

Résumé

Nous étudions la notion de temps local pour un processus continu, défini comme la limite de fonctions discrètes le long d’une suite de partitions de l’intervalle de temps. Notre approche englobe les définitions déjà existantes et coincide p.s. avec la définition (stochastique) usuelle des temps locaux pour les trajectoires de semimartingales continues. Nous établissons une version trajectorielle des formules de changement de variables, de temps et de Tanaka–Meyer ansi que plusieures conditions équivalentes pour l’existence du temps local trajectoire par trajectoire. Finalement, nous proposons une étude détaillée de la façon dont le processus limite, son temps local et sa variation quadratique, dépendent du choix de la suite de partitions. Nous montrons en particulier qu’un processus non-décroissant donné peut toujours être obtenu p.s. comme la variation quadratique d’un mouvement Brownien standard le long d’une suite appropriée de partitions (aléatoires). De tels comportements pathologiques sont cependant exclus quand les partitions sont construites à l’aide de temps d’arrêt.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 1-21.

Dates
Received: 17 December 2015
Revised: 1 September 2016
Accepted: 5 September 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030817

Digital Object Identifier
doi:10.1214/16-AIHP792

Mathematical Reviews number (MathSciNet)
MR3765878

Zentralblatt MATH identifier
06880043

Subjects
Primary: 60G17: Sample path properties 60H05: Stochastic integrals

Keywords
Pathwise local-time Itô–Tanaka formula Random partitions Brownian variation

Citation

Davis, Mark; Obłój, Jan; Siorpaes, Pietro. Pathwise stochastic calculus with local times. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 1--21. doi:10.1214/16-AIHP792. https://projecteuclid.org/euclid.aihp/1519030817


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