Deterministic Minimax Impulse Control in Finite Horizon: the Viscosity Solution Approach

TL;DR

This paper investigates a minimax impulse control problem with unbounded costs and dynamics, proving the value function's continuity and its characterization as a unique viscosity solution of an Isaacs quasi-variational inequality, with applications in finance.

Contribution

It introduces a novel approach to analyze unbounded impulse control problems using viscosity solutions, extending existing methods to more general settings.

Findings

Value function is continuous despite unbounded costs and dynamics.

Unique viscosity solution characterizes the value function.

Application demonstrated in mathematical finance.

Abstract

We study here the impulse control minimax problem. We allow the cost functionals and dynamics to be unbounded and hence the value functions can possibly be unbounded. We prove that the value function of the problem is continuous. Moreover, the value function is characterized as the unique viscosity solution of an Isaacs quasi-variational inequality. This problem is in relation with an application in mathematical finance.