# American Institute of Mathematical Sciences

October  2019, 15(4): 1965-1993. doi: 10.3934/jimo.2018132

## Optimal dividend policy with liability constraint under a hidden Markov regime-switching model

 1 School of Statistics, Faculty of Economics and Management, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China 2 Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia 3 Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, China

* Corresponding author: zjin@unimelb.edu.au

Received  March 2018 Revised  April 2018 Published  August 2018

Fund Project: The first author is supported by National Natural Science Foundation of China under Grant (Nos. 11771466, 11601157, 11231005, 11571113), Program of Shanghai Subject Chief Scientist (14XD1401600), the 111 Project (B14019). The second and third author were supported in part by Research Grants Council of the Hong Kong Special Administrative Region (project No. HKU 17330816).

This paper deals with the optimal liability and dividend strategies for an insurance company in Markov regime-switching models. The objective is to maximize the total expected discounted utility of dividend payment in the infinite time horizon in the logarithm and power utility cases, respectively. The switching process, which is interpreted by a hidden Markov chain, is not completely observable. By using the technique of the Wonham filter, the partially observed system is converted to a completely observed one and the necessary information is recovered. The upper-lower solution method is used to show the existence of classical solution of the associated second-order nonlinear Hamilton-Jacobi-Bellman equation in the two-regime case. The explicit solution of the value function is derived and the corresponding optimal dividend policies and liability ratios are obtained. In the multi-regime case, a general setting of the Wonham filter is presented, and the value function is proved to be a viscosity solution of the associated system of Hamilton-Jacobi-Bellman equations.

Citation: Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132
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