Utility maximization for reinsurance policies in a dynamic contagion claim model

Ceci Claudia, University of Rome Sapienza
Cretarola Alessandra , University of Perugia

We investigate the optimal reinsurance problem in the risk model with jump clustering features introduced in [Brachetta-Callegaro-Ceci-Sgarra, 2023]. This modeling framework is inspired by the concept initially proposed in [Dassios-Zhao, 2011] combining Hawkes and Cox processes with shot noise intensity models. Specifically, these processes describe self-exciting and externally excited jumps in the claim arrival intensity, respectively. The insurer aims to maximize the expected exponential utility of terminal wealth for general reinsurance contracts and reinsurance premia. We discuss two different methodologies: the classical stochastic control approach based on the Hamilton-Jacobi-Bellman (HJB) equation and a backward stochastic differential equation (BSDE) approach. In a Markovian setting, differently from the classical HJB-approach, the BSDE method enables us to solve the problem without imposing any requirements for regularity on the associated value function. After discussing the optimal strategy for general reinsurance contracts and reinsurance premia, we provide more explicit results in some relevant cases. Finally, by a direct computation, we establish a monotonicity property of the value function and provide comparison results that highlight the heightened risk stemming from the self-exciting component in contrast to the externally-excited counterpart. The talk is based on a joint paper with Alessandra Cretarola, University of Perugia.

Area: CS21 - Stochastic Modeling in Finance and Insurance II (Gabriele Stabile and Alessandro Milazzo)

Keywords: Optimal stochastic control; Optimal reinsurance; Hawkes and shot noise intensity Cox processes; Backward Stochastic Differential Equations (BSDEs).

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