A computational analysis of the detection problem for Brownian motion with exponential penalty based on linear programming

The quickest detection problem of a Wiener process for the case of an exponential delay-penalty was recently solved by Beibel. He derived an explicit solution to the problem exploiting the equivalence of this detection problem to an optimal stopping problem of a 2-dimensional degenerate diffusion process. In this publication we shall compute the minimal risk and the optimal stopping rule – with and without additional constraints – using linear programming models. These models are derived from a general LP approach to optimal stopping. This approach is based on a characterization of a stopped Markov process through a family of equations which relate the generator of the process to a pair of measures representing the expected occupation of the process and the distribution of the state when the process is stopped. The computational analysis of the detection problem with exponential delay-penalty leads to bounds on the minimal risk and to a range for the optimal stopping threshold. In the case of no constraints the accuracy of the numerical results will be illustrated by comparing the numerical values with the known analytical ones. While we shall prove that each constrained problem is equivalent to a particular unconstrained detection problem this correspondence does not lead to an analytical characterization of the optimal stopping rule of such problems. We shall thus computationally analyze the constrained detection problems using the aforementioned LP technique and compute optimal values and optimal stopping thresholds.