Abstract
We study the optimal excess-of-loss reinsurance problem when both the intensity of the claims arrival process and the claim size distribution are influenced by an exogenous stochastic factor. We assume that the insurer’s surplus is governed by a marked point process with dual-predictable projection affected by an environmental factor and that the insurance company can borrow and invest money at a constant real-valued risk-free interest rate r. Our model allows for stochastic risk premia, which take into account risk fluctuations. Using stochastic control theory based on the Hamilton-Jacobi-Bellman equation, we analyze the optimal reinsurance strategy under the criterion of maximizing the expected exponential utility of the terminal wealth. A verification theorem for the value function in terms of classical solutions of a backward partial differential equation is provided. Finally, some numerical results are discussed.
Highlights
In this paper, we analyze the optimal excess-of-loss reinsurance problem from the insurer’s point of view, under the criterion of maximizing the expected utility of the terminal wealth
A verification theorem for the value function in terms of classical solutions of a backward partial differential equation is provided
The main goal of our work is to extend the classical risk model by modelling the claims arrival process as a marked point process with dual-predictable projection affected by an exogenous stochastic process Y
Summary
We analyze the optimal excess-of-loss reinsurance problem from the insurer’s point of view, under the criterion of maximizing the expected utility of the terminal wealth. Among the most common arrangements, the proportional and the excess-of-loss contracts are of great interest The former was intensively studied in Irgens and Paulsen (2004); Liu and Ma (2009); Liang et al (2011); Liang and Bayraktar (2014); Zhu et al (2015); Brachetta and Ceci (2019) and references therein. The main goal of our work is to extend the classical risk model by modelling the claims arrival process as a marked point process with dual-predictable projection affected by an exogenous stochastic process Y Both the intensity of the claims arrival process and the claim size distribution are influenced by Y. Brachetta and Ceci (2019), where the authors studied the optimal proportional reinsurance In the former, the authors considered a Markov-modulated compound Poisson process, with the (unobservable) stochastic factor described by a finite state Markov chain. The paper is organized as follows: in Section 2, we formulate the model assumptions and describe the maximization problem; in Section 3 we derive the Hamilton-Jacobi-Bellman (HJB) equation; in Section 4, we investigate the candidate optimal strategy, which is suggested by the HJB derivation; in Section 5, we provide the verification argument with a probabilistic representation of the value function; in Section 6 we perform some numerical simulations
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