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.
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.
Mots clés : 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}, pages = {104--116}, publisher = {Gauthier-Villars}, volume = {45}, number = {1}, year = {2009}, doi = {10.1214/07-AIHP151}, mrnumber = {2500230}, zbl = {1171.60013}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP151/} }
TY - JOUR AU - Andres, Sebastian TI - Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 104 EP - 116 VL - 45 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP151/ DO - 10.1214/07-AIHP151 LA - en ID - AIHPB_2009__45_1_104_0 ER -
%0 Journal Article %A Andres, Sebastian %T Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 104-116 %V 45 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP151/ %R 10.1214/07-AIHP151 %G en %F 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, Tome 45 (2009) no. 1, pp. 104-116. doi : 10.1214/07-AIHP151. http://www.numdam.org/articles/10.1214/07-AIHP151/
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