Andres, Sebastian Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection. (English) Zbl 1171.60013 Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 1, 104-116 (2009). Summary: 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. Cited in 5 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J55 Local time and additive functionals 60J50 Boundary theory for Markov processes Keywords:stochastic differential equations with reflection; oblique reflection; polyhedral domains × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] H. Airault. Perturbations singulières et solutions stochastiques de problèmes de D. Neumann-Spencer. J. Math. pures et appl. 55 (1976) 233-268. · Zbl 0349.60062 [2] K. Burdzy and Z.-Q. Chen. Coalescence of synchronous couplings. Probab. Theory Related Fields 123 (2002) 553-578. · Zbl 1004.60080 · doi:10.1007/s004400200202 [3] K. L. Chung and R. J. Williams. Introduction to Stochastic Integration , 2nd edition. Birkhäuser, Boston, 1990. · Zbl 0725.60050 [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. · Zbl 1071.60044 · doi:10.1016/j.spa.2005.01.002 [5] P. Dupuis and H. Ishii. On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics 35 (1991) 31-62. · Zbl 0721.60062 [6] P. Dupuis and H. Ishii. SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 (1993) 554-580. · Zbl 0787.60099 · doi:10.1214/aop/1176989415 [7] P. L. Lions and A. S. Sznitman. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511-537. · Zbl 0598.60060 · doi:10.1002/cpa.3160370408 [8] A. Mandelbaum and K. Ramanan. Directional derivatives of oblique reflection maps. Preprint, 2005. · Zbl 1220.60054 [9] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion , 3rd edition. Springer, Heidelberg, 2005. · Zbl 1087.60040 [10] L. C. G. Rogers and D. Williams. Diffusions , Markov Processes and Martingales , Vol. 2. Cambridge Univ. Press, 2000. · Zbl 0977.60005 [11] H. Tanaka. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979) 163-177. · Zbl 0423.60055 [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. · Zbl 0579.60082 · doi:10.1002/cpa.3160380405 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.