Optimality of two-parameter strategies in stochastic control

TL;DR

This paper investigates a class of stochastic control problems where optimal strategies are characterized by two parameters, using scale functions for efficient analysis, especially in spectrally one-sided Levy process settings.

Contribution

It introduces a method to determine optimal two-parameter strategies in stochastic control problems using scale functions and verification, applicable to singular control, impulse control, and stochastic games.

Findings

Optimal strategies characterized by two parameters using smooth fit conditions.

Efficient computation of strategies via scale functions in Levy processes.

Explicit expressions for value functions in several examples.

Abstract

In this note, we study a class of stochastic control problems where the optimal strategies are described by two parameters. These include a subset of singular control, impulse control, and two-player stochastic games. The parameters are first chosen by the two continuous/smooth fit conditions, and then the optimality of the corresponding strategy is shown by verification arguments. Under the setting driven by a spectrally one-sided Levy process, these procedures can be efficiently done thanks to the recent developments of scale functions. In this note, we illustrate these techniques using several examples where the optimal strategy as well as the value function can be concisely expressed via scale functions.