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Optimal proportional reinsurance with common shock dependence to minimise the probability of drawdown

Published online by Cambridge University Press:  30 July 2018

Xia Han ,
Zhibin Liang Open the ORCID record for Zhibin Liang [Opens in a new window]  and
Caibin Zhang
Xia Han
Affiliation:
School of Mathematical Sciences, Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, P.R.China
Zhibin Liang*
Affiliation:
School of Mathematical Sciences, Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, P.R.China
Caibin Zhang
Affiliation:
School of Mathematical Sciences, Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, P.R.China
*
*Correspondence to: Zhibin Liang, School of Mathematical Sciences, Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, P.R.China. Tel: +86 18951891256. E-mail: liangzhibin111@hotmail.com
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Abstract

In this paper, we study the optimal proportional reinsurance problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component, and the criterion is to minimise the probability of drawdown, namely, the probability that the value of the surplus process reaches some fixed proportion of its maximum value to date. By the method of maximising the ratio of drift of a diffusion divided to its volatility squared, and the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman equation, we investigate the optimisation problem in two different cases. Furthermore, we constrain the reinsurance proportion in the interval [0,1] for each case, and derive the explicit expressions of the optimal proportional reinsurance strategy and the minimum probability of drawdown. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.

Keywords

Proportional reinsuranceCommon shock dependenceStochastic optimal controlProbability of drawdown
Type
Paper
Information
Annals of Actuarial Science , Volume 13 , Issue 2 , September 2019 , pp. 268 - 294
Copyright
© Institute and Faculty of Actuaries 2018 

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