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Optimal Dynamic Risk Control for Insurers with State-Dependent Income

Part of: Stochastic analysis Stochastic systems and control Mathematical economics

Published online by Cambridge University Press:  19 February 2016

Ming Zhou  and
Jun Cai
Ming Zhou*
Affiliation:
Central University of Finance and Economics
Jun Cai*
Affiliation:
University of Waterloo
*
∗ Postal address: China Institute for Actuarial Science, Central University of Finance and Economics, 39 South College Road, Haidian, Beijing 100081, China. Email address: mzhou.act@gmail.com.
∗∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. Email address: jcai@math.uwaterloo.ca.
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Abstract

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In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

Keywords

Absolute ruinexcess-of-loss reinsuranceHJB equationliquid reservestate-dependent income

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 91B30: Risk theory, insurance 91B70: Stochastic models 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 417 - 435
Copyright
© Applied Probability Trust 

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