Dynamic Reinsurance in Discrete Time Minimizing the Insurer's Cost of Capital

TL;DR

This paper extends static reinsurance optimization to a dynamic, stochastic, discrete-time setting using risk-sensitive Markov Decision Processes, deriving optimal policies under general risk measures and premium principles.

Contribution

It introduces a dynamic reinsurance model with stochastic claims and premiums, deriving the Bellman equation and proving the existence of stationary optimal policies.

Findings

Optimal reinsurance policies can be explicitly determined in examples.

The model guarantees the existence of stationary policies under infinite horizon.

The approach generalizes classical static models to dynamic, risk-sensitive settings.

Abstract

In the classical static optimal reinsurance problem, the cost of capital for the insurer's risk exposure determined by a monetary risk measure is minimized over the class of reinsurance treaties represented by increasing Lipschitz retained loss functions. In this paper, we consider a dynamic extension of this reinsurance problem in discrete time which can be viewed as a risk-sensitive Markov Decision Process. The model allows for both insurance claims and premium income to be stochastic and operates with general risk measures and premium principles. We derive the Bellman equation and show the existence of a Markovian optimal reinsurance policy. Under an infinite planning horizon, the model is shown to be contractive and the optimal reinsurance policy to be stationary. The results are illustrated with examples where the optimal policy can be determined explicitly.