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The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations
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  • Published: March 1992

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

  • C. Costantini1 

Probability Theory and Related Fields volume 91, pages 43–70 (1992)Cite this article

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Summary

The Skorohod oblique reflection problem for (D, Γ, w) (D a general domain in ℝd, Γ(x),x∈∂D, a convex cone of directions of reflection,w a function inD(ℝ+,ℝd)) is considered. It is first proved, under a condition on (D, Γ), corresponding to Γ(x) not being simultaneously too large and too much skewed with respect to ∂D, that given a sequence {w n} of functions converging in the Skorohod topology tow, any sequence {(x n, ϕn)} of solutions to the Skorohod problem for (D, Γ, w n) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, Γ, w). Next it is shown that if (D, Γ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, Γ, w) exist for everyw∈D(ℝ+,ℝd) with small enough jump size. The requirement is met in the case when ∂D is piecewiseC 1 b , Γ is generated by continuous vector fields on the faces ofD and Γ(x) makes and angle (in a suitable sense) of less than π/2 with the cone of inward normals atD, for everyx∈∂D. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.

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Authors and Affiliations

  1. Dipartimento di Matematica, Universita' di Roma “La Sapienza”, P. le A. Moro, 2, 1-00185, Roma, Italy

    C. Costantini

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  1. C. Costantini
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Costantini, C. The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations. Probab. Th. Rel. Fields 91, 43–70 (1992). https://doi.org/10.1007/BF01194489

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  • Received: 07 February 1990

  • Revised: 18 June 1991

  • Issue Date: March 1992

  • DOI: https://doi.org/10.1007/BF01194489

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Keywords

  • Brownian Motion
  • Weak Solution
  • Mathematical Biology
  • Limit Point
  • Stochastic Differential Equation
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