A multidimensional problem of optimal dividends with irreversible switching: a convergent numerical scheme

TL;DR

This paper addresses a complex multidimensional optimal dividend problem involving irreversible regime switches, providing a convergent numerical scheme to approximate the value function and illustrating it with a merger timing example.

Contribution

It introduces a novel numerical method for approximating the multidimensional optimal dividend value function with regime switches, proven to converge.

Findings

Numerical scheme converges to the optimal value function.

Method effectively handles multidimensional regime-switching control problems.

Application demonstrated through optimal merger timing for two companies.

Abstract

In this paper we study the problem of optimal dividend payment strategy which maximizes the expected discounted sum of dividends to a multidimensional set up of n associated insurance companies where the surplus process follows an n-dimensional compound Poisson process. The general manager of the companies has the possibility at any time to exercise an irreversible switch into another regime; we also take into account an expected discounted value at ruin. This multidimensional dividend problem is a mixed singular control/optimal problem. We prove that the optimal value function is a viscosity solution of the associated HJB equation and that it can be characterized as the smallest viscosity supersolution. The main contribution of the paper is to provide a numerical method to approximate (locally uniformly) the optimal value function by an increasing sequence of sub-optimal value functions of admissible strategies defined in an n-dimensional grid. As a numerical example, we present the optimal time of merger for two insurance companies.