Games of singular control and stopping driven by spectrally one-sided Levy processes

TL;DR

This paper analyzes a zero-sum game involving a spectrally one-sided Levy process, where a singular controller and stopper have opposing objectives, deriving the saddle point and value function using fluctuation theory.

Contribution

It introduces a novel game framework with spectrally one-sided Levy processes, deriving explicit solutions using scale functions and providing numerical examples.

Findings

Derived saddle point and value function for the game.

Established explicit solutions using fluctuation theory.

Provided numerical examples for phase-type Levy processes.

Abstract

We study a zero-sum game where the evolution of a spectrally one-sided Levy process is modified by a singular controller and is terminated by the stopper. The singular controller minimizes the expected values of running, controlling and terminal costs while the stopper maximizes them. Using fluctuation theory and scale functions, we derive a saddle point and the value function of the game. Numerical examples under phase-type Levy processes are also given.