Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions

Ioannis Karatzas and Johannes Ruf

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Abstract

We study one-dimensional stochastic integral equations with non-smooth dispersion coëfficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions of certain ordinary integral equations, indexed by a generic element of the underlying probability space. This relation allows us to solve the stochastic integral equations in a pathwise sense.

Résumé

Nous étudions des équations intégrales stochastiques unidimensionnelles avec coefficient de diffusion non-régulier, et avec termes de dérive non nécessairement absolument continus par rapport à la mesure de Lebesgue. En s’inspirant de Lamperti, Doss et Sussmann, la résolution de ces équations se ramène à la résolution de certaines équations intégrales ordinaires, paramétrées par un élément $\omega$ variant dans l’espace de probabilité de base. Ce lien nous permet de résoudre les équations intégrales stochastiques d’une façon trajectorielle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 915-938.

Dates
Received: 13 December 2013
Revised: 16 July 2014
Accepted: 28 October 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367900

Digital Object Identifier
doi:10.1214/14-AIHP660

Mathematical Reviews number (MathSciNet)
MR3498016

Zentralblatt MATH identifier
1346.45010

Subjects
Primary: 34A99: None of the above, but in this section 45J05: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20] 60G48: Generalizations of martingales 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Stochastic integral equation Ordinary integral equation Pathwise solvability Existence Uniqueness Generalized drift Wong–Zakai approximation Support theorem Comparison theorem Stratonovich integral

Citation

Karatzas, Ioannis; Ruf, Johannes. Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 915--938. doi:10.1214/14-AIHP660. https://projecteuclid.org/euclid.aihp/1462367900


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