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Explicit Solutions for Variational Problems in the Quadrant

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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 R d +. 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.

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Authors and Affiliations

  1. Department of Statistics and Actuarial Science, Heriot Watt University, Edinburgh, Scotland

    F. Avram

  2. School of Industrial and Systems Engineering and School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0205, USA

    J.G. Dai

  3. Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, University of Texas at Austin, Austin, TX, 78712-1063, USA

    J.J. Hasenbein

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  1. F. Avram
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  2. J.G. Dai
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  3. J.J. Hasenbein
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Avram, F., Dai, J. & Hasenbein, J. Explicit Solutions for Variational Problems in the Quadrant. Queueing Systems 37, 259–289 (2001). https://doi.org/10.1023/A:1011004620420

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