Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, p. 104-116

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.

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.

DOI : https://doi.org/10.1214/07-AIHP151
Classification:  60H10,  60J55,  60J50
Keywords: stochastic differential equations with reflection, oblique reflection, polyhedral domains
@article{AIHPB_2009__45_1_104_0,
     author = {Andres, Sebastian},
     title = {Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     pages = {104-116},
     doi = {10.1214/07-AIHP151},
     zbl = {1171.60013},
     mrnumber = {2500230},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_1_104_0}
}
Andres, Sebastian. Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 104-116. doi : 10.1214/07-AIHP151. http://www.numdam.org/item/AIHPB_2009__45_1_104_0/

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