Value Function and Optimal Rule on the Optimal Stopping Problem for Continuous-Time Markov Processes

Lu Ye

doi:10.1155/2017/3596037

Abstract

This paper considers the optimal stopping problem for continuous-time Markov processes. We describe the methodology and solve the optimal stopping problem for a broad class of reward functions. Moreover, we illustrate the outcomes by some typical Markov processes including diffusion and Lévy processes with jumps. For each of the processes, the explicit formula for value function and optimal stopping time is derived. Furthermore, we relate the derived optimal rules to some other optimal problems.

Highlights

IntroductionLet (Ω, F, P) be a complete probability space; the problem studied in this paper is to find the optimum
Let (Ω, F, P) be a complete probability space; the problem studied in this paper is to find the optimum V (x) = sup τEx [e−r(τ−t)f (Xτ) | Ft] (1)where r > 0 is the discount rate and τ ≥ t is a stopping time
We show the conditions for reward functions and deduce the explicit optimal rules for general continuous-time Markov processes including diffusion and Levy processes with jumps

Summary

Introduction

Let (Ω, F, P) be a complete probability space; the problem studied in this paper is to find the optimum. We show the conditions for reward functions and deduce the explicit optimal rules for general continuous-time Markov processes including diffusion and Levy processes with jumps. Our work naturally explore the explicit solutions of the new optimal problem (4) for a larger class of reward functions and underlying processes Xt. The paper is organized as follows.

Optimal Rule for Continuous-Time Markov Processes

Diffusion

Lévy Processes with Jumps

Connection with the Other Optimal Problems

Conclusions