Optimal stopping for Markov processes with positive jumps

TL;DR

This paper characterizes the optimal stopping region for Markov processes with positive jumps, providing a threshold-based criterion and a formula for the value function, with applications to Levy-driven Ornstein-Uhlenbeck processes in electricity markets.

Contribution

It offers a new verification theorem for the structure of the optimal stopping region and a representation formula for the value function using the Green kernel, applied to Levy-driven processes.

Findings

Optimal stopping region is {x >= x^*} for some threshold x^*

Representation formula for the value function via Green kernel

Application to electricity market price modeling

Abstract

Consider the discounted optimal stopping problem for a real valued Markov process with only positive jumps. We provide a theorem to verify that the optimal stopping region has the form {x >= x^*} for some critical threshold x^*, and a representation formula for the value function of the problem in terms of the Green kernel of the process, based on Dynkin's characterization of the value function as the least excessive majorant. As an application of our results, using the Fourier transform to compute the Green kernel of the process, we solve a new example: the optimal stopping for a Levy-driven Ornstein-Uhlenbeck process used to model prices in electricity markets.