# The Finite Horizon impulse control Problem with arbitrary cost functions : the Viscosity Solution Approach

## TL;DR

This paper addresses stochastic impulse control problems with arbitrary cost functions, demonstrating that the value function uniquely solves the associated HJB PDE using viscosity solutions, thus extending the theoretical framework for such control problems.

## Contribution

It establishes the uniqueness of the viscosity solution for the HJB equation in impulse control with arbitrary costs, broadening the applicability of the viscosity solutions approach.

## Key findings

- Proves the value function is a unique viscosity solution.
- Extends the viscosity solutions framework to arbitrary cost functions.
- Provides a rigorous mathematical foundation for stochastic impulse control problems.

## Abstract

We consider stochastic impulse control problems when the impulses cost functions are arbitrary. We use the dynamic programming principle and viscosity solutions approach to show that the value function is a unique viscosity solution for the associated Hamilton-Jacobi-Bellman equation (HJB) partial differential equation (PDE) of stochastic impulse control problems