On the Solution of General Impulse Control Problems Using Superharmonic Functions

TL;DR

This paper characterizes solutions to general impulse control problems using superharmonic functions within a Markovian framework, providing a theoretical foundation for deriving optimal strategies and illustrating with diverse examples.

Contribution

It introduces a novel characterization of impulse control solutions as minimal superharmonic functions, extending the theory to a broad class of problems.

Findings

Value function is the minimal superharmonic function in a convex set.

Provides a method to derive optimal impulse control strategies.

Illustrates the theory with examples from multiple stopping and switching problems.

Abstract

In this paper, a characterization of the solution of impulse control problems in terms of superharmonic functions is given. In a general Markovian framework, the value function of the impulse control problem is shown to be the minimal function in a convex set of superharmonic functions. This characterization also leads to optimal impulse control strategies and can be seen as the corresponding characterization to the description of the value function for optimal stopping problems as a smallest superharmonic majorant of the reward function. The results are illustrated with examples from different fields, including multiple stopping and optimal switching problems.