Optimal control of risk process in a regime-switching environment

TL;DR

This paper develops an optimal control framework for an insurance company's surplus modeled by a regime-switching diffusion, aiming to minimize costs until exit, with new conditions ensuring the value function's continuity and viscosity solution properties.

Contribution

It introduces a weaker sufficient condition for the value function's continuity and establishes its characterization as a viscosity solution of the Hamilton-Jacobian-Bellman equation.

Findings

Derived a weaker condition for value function continuity.

Proved the value function is a viscosity solution.

Applied the model to optimize insurance risk management.

Abstract

This paper is concerned with cost optimization of an insurance company. The surplus of the insurance company is modeled by a controlled regime switching diffusion, where the regime switching mechanism provides the fluctuations of the random environment. The goal is to find an optimal control that minimizes the total cost up to a stochastic exit time. A weaker sufficient condition than that of (Fleming and Soner 2006, Section V.2) for the continuity of the value function is obtained. Further, the value function is shown to be a viscosity solution of a Hamilton-Jacobian-Bellman equation.