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.

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.

DOI : 10.1214/07-AIHP151
Classification : 60H10, 60J55, 60J50
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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1214/07-AIHP151/

[1] H. Airault. Perturbations singulières et solutions stochastiques de problèmes de D. Neumann-Spencer. J. Math. pures et appl. 55 (1976) 233-268. | MR | Zbl

[2] K. Burdzy and Z.-Q. Chen. Coalescence of synchronous couplings. Probab. Theory Related Fields 123 (2002) 553-578. | MR | Zbl

[3] K. L. Chung and R. J. Williams. Introduction to Stochastic Integration, 2nd edition. Birkhäuser, Boston, 1990. | MR | Zbl

[4] J.-D. Deuschel and L. Zambotti. Bismut-Elworthy formula and random walk representation for SDEs with reflection. Stochastic Process. Appl. 115 (2005) 907-925. | MR | Zbl

[5] P. Dupuis and H. Ishii. On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics 35 (1991) 31-62. | MR | Zbl

[6] P. Dupuis and H. Ishii. SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 (1993) 554-580. | MR | Zbl

[7] P. L. Lions and A. S. Sznitman. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511-537. | MR | Zbl

[8] A. Mandelbaum and K. Ramanan. Directional derivatives of oblique reflection maps. Preprint, 2005.

[9] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Heidelberg, 2005. | Zbl

[10] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales, Vol. 2. Cambridge Univ. Press, 2000. | MR | Zbl

[11] H. Tanaka. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979) 163-177. | MR | Zbl

[12] S. R. S. Varadhan and R. J. Williams. Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 (1984) 405-443. | MR | Zbl

Cité par Sources :