De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process

TL;DR

This paper explores the joint dividend payout strategies for two interconnected insurance portfolios modeled as a two-dimensional risk process, analyzing how dependencies influence optimal control policies and the timing of dividends.

Contribution

It introduces the first analysis of joint dividend strategies in a two-dimensional risk model with dependencies, contrasting with traditional one-dimensional models.

Findings

Optimal dividend barriers depend on initial reserves in the two-dimensional model.

Dependency between portfolios affects the timing and size of dividend payouts.

The model distinguishes between refraction and impulse control scenarios.

Abstract

Consider two insurance companies (or two branches of the same company) that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal). We model the occurrence of claims according to a Poisson process. The ruin is achieved when the corresponding two-dimensional risk process first leaves the positive quadrant. We will consider two scenarios of the controlled process: refraction and impulse control. In the first case the dividends are payed out when the two-dimensional risk process exits the fixed region. In the second scenario, whenever the process hits the horizontal line, it is reduced by paying dividends to some fixed point in the positive quadrant where it waits for the next claim to arrive. In both models we calculate the discounted cumulative dividend payments until the ruin. This paper is the first attempt to understand the effect of dependencies of two portfolios on the joint optimal strategy of paying dividends. For example in case of proportional reinsurance one can observe the interesting phenomenon that choice of the optimal barrier depends on the initial reserves. This is in contrast with the one-dimensional Cram\'{e}r-Lundberg model where the optimal choice of the barrier is uniform for all initial reserves.