Optimal Dividend and Investment Problems under Sparre Andersen Model

TL;DR

This paper addresses optimal dividend and investment strategies when the reserve process follows a non-Markovian Sparre Andersen model, using backward Markovization to analyze the problem within a Markovian framework.

Contribution

It introduces a Markovian reformulation of the non-Markovian Sparre Andersen model for optimal control, establishing regularity and uniqueness of the value function as a viscosity solution.

Findings

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

The Markovian reformulation enables analysis of the non-Markovian reserve process.

The approach validates the dynamic programming principle for the model.

Abstract

In this paper we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a "renewal" process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution} to the associated HJB equation on a cylindrical domain on which the problem is well-defined.