Optimal dividends for a two-dimensional risk model with simultaneous ruin of both branches

TL;DR

This paper addresses the optimal dividend distribution problem in a two-dimensional risk model where both branches can simultaneously face ruin, deriving the HJB equation and identifying the optimal strategy.

Contribution

It formulates the stochastic control problem for a degenerate bivariate risk model, proves the viscosity solution property, and reduces the control problem to a simpler one-dimensional case.

Findings

Optimal value function satisfies a specific HJB equation.

The value function is the smallest viscosity solution with growth conditions.

Numerical examples illustrate the theoretical results.

Abstract

We consider the optimal dividend problem in the so-called degenerate bivariate risk model under the assumption that the surplus of one branch may become negative. More specific, we solve the stochastic control problem of maximizing discounted dividends until simultaneous ruin of both branches of an insurance company by showing that the optimal value function satisfies a certain Hamilton-Jacobi-Bellman (HJB) equation. Further, we prove that the optimal value function is the smallest viscosity solution of said HJB equation, satisfying certain growth conditions. Under some additional assumptions, we show that the optimal strategy lies within a certain subclass of all admissible strategies and reduce the two-dimensional control problem to a one-dimensional one. The results are illustrated by a numerical example and Monte-Carlo simulated value functions.