Optimal Claim-Dependent Proportional Reinsurance Under a Self-Exciting Claim Model

Fan Wu (), Yang Shen (), Xin Zhang () and Kai Ding ()
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Fan Wu: Southeast University
Yang Shen: University of New South Wales
Xin Zhang: Southeast University
Kai Ding: Southeast University

Journal of Optimization Theory and Applications, 2024, vol. 201, issue 3, No 10, 1229-1255

Abstract: Abstract This paper investigates an optimal reinsurance problem for an insurance company with self-exciting claims, where the insurer’s historical claims affect the claim intensity itself. We focus on a claim-dependent proportional reinsurance contact, where the term “claim-dependent” signifies that the insurer’s risk retention ratio is allowed to depend on claim size. The insurer aims to maximize the expected utility of terminal wealth. By utilizing the dynamic programming principle and verification theorem, we obtain the optimal reinsurance strategy and corresponding value function in closed-form from the Hamilton–Jacobi–Bellman equation under an exponential utility function. We show that the claim-dependent proportional reinsurance is optimal among all types of reinsurance under the exponential utility maximization criterion. In addition, we present several analytical properties and numerical examples of the derived optimal strategy and provide economic insights through analytical and numerical analyses. In particular, we show the optimal claim-dependent proportional reinsurance can be considered as a continuous approximation of the step-wise risk sharing rule between the insurer and the reinsurer.

Keywords: Optimal reinsurance problem; Self-exciting claim; Claim-dependent proportional reinsurance; Dynamic programming principle; Verification theorem; 49L20; 49N90; 91G80 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02429-y

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