Explicit Solutions for Variational Problems in the Quadrant

Authors:
F. Avram;J. G. Dai;J. J. Hasenbein
Affiliations:
Department of Statistics and Actuarial Science, Heriot Watt University, Edinburgh, Scotland F.Avram@ma.hw.ac.uk;School of Industrial and Systems Engineering and School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA dai@isye.gatech.edu;Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, University of Texas at Austin, Austin, TX 78712-1063, USA jhas@mail.utexas.edu
Venue:
Queueing Systems: Theory and Applications
Year:
2001

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Abstract

We study a variational problem (VP) that is related to semimartingale reflecting Brownian motions (SRBMs). Specifically, this VP appears in the large deviations analysis of the stationary distribution of SRBMs in the d-dimensional orthant Rd+. When d=2, we provide an explicit analytical solution to the VP. This solution gives an appealing characterization of the optimal path to a given point in the quadrant and also provides an explicit expression for the optimal value of the VP. For each boundary of the quadrant, we construct a “cone of boundary influence”, which determines the nature of optimal paths in different regions of the quadrant. In addition to providing a complete solution in the 2-dimensional case, our analysis provides several results which may be used in analyzing the VP in higher dimensions and more general state spaces.