Compound Poisson disorder with general prior and misspecified Wiener disorder problem

Abstract

For a system modeled with a compound Poisson or a Wiener process, let us assume that the underlying model parameters change at an unknown and unobservable time. For a compound Poisson process, these are arrival rate and mark distribution while for a Wiener process, it is the drift parameter. Suppose the decision maker knows the pre- and post-disorder process parameters, as well as the prior density of the disorder time. In this case, finding a stopping rule that optimizes a Bayesian penalty function is called the compound Poisson and Wiener disorder problem, respectively. For the compound Poisson problem, we consider a general prior distribution where the decision maker has more general knowledge about the disorder time than exponential and uniform priors which were addressed in the previous studies. For the Wiener problem, we revisit the asset selling problem with an exponential prior, where the decision maker specifies problem parameters incorrectly. In both cases, the original problems reduce to optimal stopping problems. We use time discretization and successive approximation methods for the first case and Markov chain approximation and Monte Carlo simulations for the second case. We provide the quickest detection rules and discuss various numerical examples.