Stochastic Optimal Impulse Controls with Changing Running Costs

TL;DR

This paper develops a framework for stochastic impulse control problems with costs that depend on the impulses, deriving the associated HJB equation and establishing a maximum principle for optimal impulses.

Contribution

It introduces a novel approach to impulse control with changing costs, deriving the HJB equation with a parameter and proving a maximum principle for such controls.

Findings

The value function is the unique viscosity solution to the derived HJB equation.

A maximum principle for stochastic impulse controls with perturbations in impulse timing is established.

The HJB equation is a quasi-variational inequality with a parameter.

Abstract

This paper is concerned with stochastic impulse control problems in which the running cost changes depending on the impulse control. Because of such a dependence, it brings several difficulties when the usual dynamic programming principle is to be used. The corresponding Hamilton-Jacobi-Bellman (HJB) equation (a quasi-variational inequality) is derived, which contains a parameter. The value function is a unique viscosity solution to this HJB equation by a classical argument. Further, inspired by the derivation of the Pontryagin type maximum principle for stochastic optimal controls with a non-convex control domain, we have established the maximum principle for our stochastic optimal impulse controls, allowing perturbations in optimal impulse moments.