A Characterization of the Reflected Quasipotential

A Characterization of the Reflected Quasipotential Recent interest in the reflected quasipotential comes from the queueing theory literature, specifically the analysis of so-called $$(b,A,D)$$ ( b , A , D ) reflected Brownian motion where it is the large deviation rate function for the stationary distribution. Our purpose here is to characterize the reflected quasipotential in terms of a first-order Hamilton–Jacobi equation. Using conventional dynamic programming ideas, along with a complementarity problem formulation of the effect of the Skorokhod map on absolutely continuous paths, we will derive necessary conditions in the form of viscosity-sense boundary conditions. It turns out that even with these boundary conditions solutions are not unique. Thus a unique characterization needs to refer to some additional property of $$V(\cdot )$$ V ( · ) . We establish such a characterization in two dimensions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A Characterization of the Reflected Quasipotential

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Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Science+Business Media New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-014-9286-9
Publisher site
See Article on Publisher Site

Abstract

Recent interest in the reflected quasipotential comes from the queueing theory literature, specifically the analysis of so-called $$(b,A,D)$$ ( b , A , D ) reflected Brownian motion where it is the large deviation rate function for the stationary distribution. Our purpose here is to characterize the reflected quasipotential in terms of a first-order Hamilton–Jacobi equation. Using conventional dynamic programming ideas, along with a complementarity problem formulation of the effect of the Skorokhod map on absolutely continuous paths, we will derive necessary conditions in the form of viscosity-sense boundary conditions. It turns out that even with these boundary conditions solutions are not unique. Thus a unique characterization needs to refer to some additional property of $$V(\cdot )$$ V ( · ) . We establish such a characterization in two dimensions.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 1, 2015

References

  • Explicit solutions for variational problems in the quadrant
    Avram, F; Dai, JG; Hasenbein, JJ

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