Explicit Solutions for Variational Problems in the Quadrant
Florin Avram, J. G. Dai, J. J. Hasenbein
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. 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.