# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018154

## Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax

 1 School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China 2 School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China 3 School of Economics, Nanjing University of Finance and Economics, Nanjing 210023, China

Received  March 2018 Revised  May 2018 Published  September 2018

This study examines the optimal investment and dividend problem for an insurer with CRRA preference. The insurer's goal is to maximize the expected discounted accumulated utility from dividend before ruin and the insurer subjects to high gain tax payment. Both the surplus process and the financial market are modulated by an external Markov chain. Using the weak dynamic programming principle (WDPP), we prove that the value function of our control problem is the unique viscosity solution to coupled Hamilton-Jacobi-Bellman (HJB) equations with first derivative constraints. Solving an auxiliary problem without regime switching, we prove that, it is optimal for the insurer in a multiple-regime market to adopt the policies in the same way as in a single-regime market. The regularity of the viscosity solution on its domain is proved and thus the HJB equations admits classical solution. A numerical scheme for the value function is provided by the Markov chain approximation method, two numerical examples are given to illustrate the impact of the high gain tax and regime switching on the optimal policies.

Citation: Lin Xu, Dingjun Yao, Gongpin Cheng. Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018154
##### References:

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##### References:
Comparison of optimal dividend amount
Comparison of optimal investment amount
Optimal dividend amount under bull and bear market
Optimal investment amount under bull and bear marke
Parameters setting in Example 1
 (Investor, Parameter) $\mu_1$ $\sigma_1$ $\mu_2$ $\sigma_2$ $p$ $\beta$ $\lambda$ $n$ Merton 0 0 0.06 0.3 0.33 0.15 0 - Financial Agent 0 0 0.06 0.3 0.33 0.15 0.2 10 Insurer 0.4 0.5 0.06 0.3 0.33 0.15 0.2 10
 (Investor, Parameter) $\mu_1$ $\sigma_1$ $\mu_2$ $\sigma_2$ $p$ $\beta$ $\lambda$ $n$ Merton 0 0 0.06 0.3 0.33 0.15 0 - Financial Agent 0 0 0.06 0.3 0.33 0.15 0.2 10 Insurer 0.4 0.5 0.06 0.3 0.33 0.15 0.2 10
Parameters setting in Example 2
 (State, Parameter) $\mu_1$ $\sigma_1$ $\mu_2$ $\sigma_2$ $p$ $\beta$ $\lambda$ $n$ Bull 0.4 0.5 0.06 0.3 0.3 0.05 0.2 10 Bear 0.3 0.5 0.03 0.3 0.3 0.05 0.2 10
 (State, Parameter) $\mu_1$ $\sigma_1$ $\mu_2$ $\sigma_2$ $p$ $\beta$ $\lambda$ $n$ Bull 0.4 0.5 0.06 0.3 0.3 0.05 0.2 10 Bear 0.3 0.5 0.03 0.3 0.3 0.05 0.2 10
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