# American Institute of Mathematical Sciences

September  2019, 9(3): 425-452. doi: 10.3934/mcrf.2019020

## A fully nonlinear free boundary problem arising from optimal dividend and risk control model

 1 School of Mathematics, Jiaying University, Meizhou 514015, China 2 School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China 3 School of Mathematical Science, South China Normal University, Guangzhou 510631, China

* Corresponding author

Received  March 2017 Revised  October 2018 Published  April 2019

Fund Project: The first author is supported by NNSF of China (No.11626117 and No.11601163), NSF of Guangdong Province of China (No.2016A030307008). The second author is supported by NNSF of China (No.11771158 and No.71871071), NSF of Guangdong Province of China (No.2016A030313448, No.2017A030313397 and No.2018B030311004).

Focusing on the problem arising from a stochastic model of risk control and dividend optimization techniques for a financial corporation, this work considers a parabolic variational inequality with gradient constraint
 $\min\Big\{v_t-\max\limits_{0\leq a\leq1}\Big(\frac{1}{2}\sigma^2a^2v_{xx}+\mu av_x\Big)+cv,\;v_x-1\Big\} = 0.$
Suppose the company's performance index is the total discounted expected dividends, our objective is to choose a pair of control variables so as to maximize the company's performance index, which is the solution to the above variational inequality under certain initial-boundary conditions. The main effort is to analyse the properties of the solution and two free boundaries arising from the above variational inequality, which we call dividend boundary and reinsurance boundary.
Citation: Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control & Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020
##### References:

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##### References:
Penalty function
Dividend free boundary
Reinsurance free boundary
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