Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection

Sebastian Andres

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 1, page 104-116
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Andres, Sebastian. "Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 104-116. <http://eudml.org/doc/78010>.

@article{Andres2009,
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.},
author = {Andres, Sebastian},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic differential equations with reflection; oblique reflection; polyhedral domains},
language = {eng},
number = {1},
pages = {104-116},
publisher = {Gauthier-Villars},
title = {Pathwise differentiability for SDEs in a convex polyhedron with oblique reflection},
url = {http://eudml.org/doc/78010},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 1
SP - 104
EP - 116
AB - 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.
LA - eng
KW - stochastic differential equations with reflection; oblique reflection; polyhedral domains
UR - http://eudml.org/doc/78010
ER -

References

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  6. [6] P. Dupuis and H. Ishii. SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 (1993) 554–580. Zbl0787.60099MR1207237
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  8. [8] A. Mandelbaum and K. Ramanan. Directional derivatives of oblique reflection maps. Preprint, 2005. 
  9. [9] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Heidelberg, 2005. Zbl1087.60040
  10. [10] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales, Vol. 2. Cambridge Univ. Press, 2000. Zbl0977.60005MR1780932
  11. [11] H. Tanaka. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979) 163–177. Zbl0423.60055MR529332
  12. [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. Zbl0579.60082MR792398

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