Minimax Impulse Control Problems in Finite Horizon

TL;DR

This paper studies finite-horizon minimax impulse control problems, establishing the existence and uniqueness of the value function as a viscosity solution to an Isaacs quasi-variational inequality, with applications in finance.

Contribution

It proves the existence and uniqueness of the value function for impulse control minimax problems and characterizes it via a viscosity solution, linking it to financial applications.

Findings

Existence of the value function is established.

The value function is the unique viscosity solution.

Application to mathematical finance is demonstrated.

Abstract

We consider the problem of impulse control minimax in finite horizon, when cost functions $(C(t,x,\xi)>0)$. We show existence of value function of the problem. 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.