Minimizing the Probability of Absolute Ruin Under Ambiguity Aversion
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Minimizing the Probability of Absolute Ruin Under Ambiguity Aversion Xia Han1 · Zhibin Liang1
· Kam Chuen Yuen2 · Yu Yuan1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider an optimal robust reinsurance problem in a diffusion model for an ambiguity-averse insurer, who worries about ambiguity and aims to minimize the robust value involving the probability of absolute ruin and a penalization of model ambiguity. It is assumed that the insurer is allowed to purchase per-claim reinsurance to transfer its risk exposure, and that the reinsurance premium is computed according to the mean-variance premium principle which is a combination of the expected-value and variance premium principles. The optimal reinsurance strategy and the associated value function are derived explicitly by applying stochastic dynamic programming and by solving the corresponding boundary-value problem. We prove that there exists a unique point of inflection which relies on the penalty parameter greatly such that the robust value function is strictly concave up to the unique point of inflection and is strictly convex afterwards. It is also interesting to observe that the expression of the optimal robust reinsurance strategy is independent of the penalty parameter and coincides with the one in the benchmark case without ambiguity. Finally, some numerical examples are presented to illustrate the effect of ambiguity aversion on our optimal results. Keywords Absolute ruin probability · Ambiguity aversion · Mean-variance premium principle · Per-claim reinsurance · Robust optimization
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Zhibin Liang [email protected] Xia Han [email protected] Kam Chuen Yuen [email protected] Yu Yuan [email protected]
1
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, Jiangsu, People’s Republic of China
2
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong
123
Applied Mathematics & Optimization
Mathematics Subject Classification 62P05 · 90C39 · 91B30 · 93E20
1 Introduction Risk models taking reinsurance into consideration have received a great deal of attention in the literature because reinsurance is an effective approach for insurers to manage their risk exposures so as to achieve their financial objectives. Subject to reinsurance control with or without investment control, optimization problems under various objective functions have become a popular research topic in the actuarial literature. Schmidli [1], Bai and Guo [2], and Liang and Young [3] considered the optimal reinsurance-investment problems in which an insurance company wishes to minimize the probability of ruin. Under the criterion of maximizing the expected utility of terminal wealth, the optimal reinsurance or investment strategies were studied in Liu and Ma [4], Liang and Bayraktar [5], and Liang and Yuen [6]. It is worth noting that the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman (HJB) equation are widely
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