Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection

Sebastian Andres

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Abstract

In this paper, the object of study is a Skorohod SDE in a convex polyhedron with oblique reflection at the boundary. We prove that the solution is pathwise differentiable with respect to its deterministic starting point up to the time when two of the faces are hit simultaneously. The resulting derivatives evolve according to an ordinary differential equation, when the process is in the interior of the polyhedron, and they are projected to the tangent space, when the process hits the boundary, while they jump in the direction of the corresponding reflection vector.

Résumé

L’object du présent travail est l’étude d’une équation différentielle stochastique de type Skorohod dans un polyèdre convexe avec réflexions obliques au bord. Nous démontrons que pour presque toutes les trajectoires, la solution est différentiable par rapport au point de départ jusqu’au temps où deux faces sont atteintes simultanément. Les dérivées sont à l’intérieur du polyèdre solutions d’une équation différentielle ordinaire. Au bord du polyèdre elles sont projetées dans l’espace tangeant en sautant en direction du vecteur de reflection correspondant.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 1 (2009), 104-116.

Dates
First available in Project Euclid: 12 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1234469973

Digital Object Identifier
doi:10.1214/07-AIHP151

Mathematical Reviews number (MathSciNet)
MR2500230

Zentralblatt MATH identifier
1171.60013

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J55: Local time and additive functionals 60J50: Boundary theory

Keywords
Stochastic differential equations with reflection Oblique reflection Polyhedral domains

Citation

Andres, Sebastian. Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 104--116. doi:10.1214/07-AIHP151. https://projecteuclid.org/euclid.aihp/1234469973


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