A dynamic optimal reinsurance strategy with capital injections in the Cramer-Lundberg model

TL;DR

This paper derives an explicit, optimal dynamic reinsurance and dividend strategy for an insurance surplus modeled by the Cramer-Lundberg process, incorporating capital injections and using analytical solutions to the Hamilton Jacobi Bellman equation.

Contribution

It provides a comprehensive, explicit characterization of the optimal reinsurance and dividend policy with capital injections in the Cramer-Lundberg model, including analytical methods and illustrative examples.

Findings

Explicit optimal policy characterized via Hamilton Jacobi Bellman equation

Strategy involves stopping at overshoot threshold and paying dividends at upper barrier

Examples demonstrate applicability to proportional reinsurance treaties

Abstract

In this article we consider the surplus process of an insurance company within the Cramer-Lundberg framework. We study the optimal reinsurance strategy and dividend distribution of an insurance company under proportional reinsurance, in which capital injections are allowed. Our aim is to find a general dynamic reinsurance strategy that maximises the expected discounted cumulative dividends until the time of passage below a given level, called ruin. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a, and only pay dividends when the reserve reaches an upper barrier b. Using analytical methods, we identify the value function as a particular solution to the associated Hamilton Jacobi Bellman equation. This approach leads to an exhaustive and explicit characterisation of optimal policy. The proportional reinsurance is given via comprehensive structure equations. Furthermore we give some examples illustrating the applicability of this method for proportional reinsurance treaties.