Abstract
In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and the reinsurance price is modelled by a continuous-time Markov chain with two states. The presence of regime-switching substantially complicates the optimal reinsurance problem, as the surplus-independent strategies turn out to be suboptimal. We develop a recursive approach that allows to represent a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation and the corresponding reinsurance strategy as the unique limits of the sequence of solutions to ordinary differential equations and their first- and second-order derivatives. Via Ito’s formula, we prove the constructed function to be the value function. Two examples illustrate the recursive procedure along with a numerical approach yielding the direct solution to the HJB equation.
Highlights
IntroductionWriting red numbers is generally considered a bad sign for the financial health of a (insurance) company
Writing red numbers is generally considered a bad sign for the financial health of a company
We study the problem faced by an insurance company that aims at finding the optimal proportional reinsurance strategy minimising the expected discounted capital injections
Summary
Writing red numbers is generally considered a bad sign for the financial health of a (insurance) company. One can control the capital injections—representing the company’s riskiness—by reinsurance In this context, the problem of finding a reinsurance strategy leading to a minimal possible value of expected discounted capital injections has been solved in Eisenberg and Schmidli (2009). There, the optimal reinsurance strategy is given by a constant, meaning that the insurance company is choosing a retention level once and forever This result has been obtained under the assumption that the parameters describing the evolution of the insurer’s and reinsurer’s surplus never change. If the discounting rate would be assumed to be negative in one of the states, it might become optimal to inject capital even if the surplus is still positive (see, for instance, in Eisenberg and Krühner (2018)), which would substantially complicate the problem.
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