Optimal stopping problems for some Markov processes

TL;DR

This paper explicitly solves optimal stopping problems for certain Markov processes with random discounting and additive costs, extending previous results to broader classes of processes using potential theory and stochastic calculus.

Contribution

It generalizes existing solutions of optimal stopping problems to include one-sided regular Feller processes and provides explicit solutions for a wider class of Markov processes.

Findings

Explicit solutions for optimal stopping with random discounting.

Extension to one-sided regular Feller processes.

Application to new examples including spectrally negative processes.

Abstract

In this paper, we solve explicitly the optimal stopping problem with random discounting and an additive functional as cost of observations for a regular linear diffusion. We also extend the results to the class of one-sided regular Feller processes. This generalizes the result of Beibel and Lerche [Statist. Sinica 7 (1997) 93-108] and [Teor. Veroyatn. Primen. 45 (2000) 657-669] and Irles and Paulsen [Sequential Anal. 23 (2004) 297-316]. Our approach relies on a combination of techniques borrowed from potential theory and stochastic calculus. We illustrate our results by detailing some new examples ranging from linear diffusions to Markov processes of the spectrally negative type.