# 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:
 [1] X. Chen, Y. Chen and F. Yi, Parabolic variational inequality with parameter and gradient constraints, J. Math. Anal. Appl., 385 (2012), 928-946.  doi: 10.1016/j.jmaa.2011.07.025.  Google Scholar [2] X. Chen and F. Yi, A problem of singular stochastic control with optimal stopping in finite horizon, SIAM J. Control Optim., 50 (2012), 2151-2172.  doi: 10.1137/110832264.  Google Scholar [3] M. Dai and F. Yi, Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem, J. Differ. Equ., 246 (2009), 1445-1469.  doi: 10.1016/j.jde.2008.11.003.  Google Scholar [4] A. Friedman, Partial Differential Equaions of Parabolic Type, Prentice-Hall Inc., 1964.  Google Scholar [5] A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Funct. Anal., 18 (1975), 151-176.  doi: 10.1016/0022-1236(75)90022-1.  Google Scholar [6] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [7] C. Guan and F. Yi, A free boundary problem arising from a stochastic optimal control model with bounded dividend rate, Stoch. Anal. Appl., 32 (2014), 742-760.  doi: 10.1080/07362994.2014.922778.  Google Scholar [8] C. Guan and F. Yi, A free boundary problem arising from a stochastic optimal control model under controllable risk, J. Differ. Equ., 260 (2016), 4845-4870.  doi: 10.1016/j.jde.2015.10.040.  Google Scholar [9] B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quant. Financ., 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007.  Google Scholar [10] D. Kelome and A. Swiech, Viscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equation, Appl. Math. Optim., 47 (2003), 253-278.  doi: 10.1007/s00245-003-0764-8.  Google Scholar [11] A. Kolesnichenko and G. Shopina, Valuation of portfolios under uncertain volatility: Black-Scholes-Barenblatt equation and the static hedging, Technical Report, IDE0739, 2007. Google Scholar [12] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar [13] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8.  Google Scholar [14] V. A. Solonnikov, O. A. Ladyzenskaja and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Rusian by Sussian by Smith, S. 1967. Translations of Mathematical Monographs, volume 23, American Mathematical Society, 1968.  Google Scholar [15] M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Math. Meth. of Oper. Res., 51 (2000), 1-42.  doi: 10.1007/s001860050001.  Google Scholar [16] M. Taksar and X. Zhou, Optimal risk and dividend control for a company with a debt liability, Insurance: Mathematics and Economics, 22 (1998), 105-122.  doi: 10.1016/S0167-6687(98)00012-2.  Google Scholar [17] T. Vargiolu, Existence, uniqueness and smoothness for the Black-Scholes-Barenblatt equation, Universita Di Padova, 4 (2001), 315-327.   Google Scholar

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##### References:
 [1] X. Chen, Y. Chen and F. Yi, Parabolic variational inequality with parameter and gradient constraints, J. Math. Anal. Appl., 385 (2012), 928-946.  doi: 10.1016/j.jmaa.2011.07.025.  Google Scholar [2] X. Chen and F. Yi, A problem of singular stochastic control with optimal stopping in finite horizon, SIAM J. Control Optim., 50 (2012), 2151-2172.  doi: 10.1137/110832264.  Google Scholar [3] M. Dai and F. Yi, Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem, J. Differ. Equ., 246 (2009), 1445-1469.  doi: 10.1016/j.jde.2008.11.003.  Google Scholar [4] A. Friedman, Partial Differential Equaions of Parabolic Type, Prentice-Hall Inc., 1964.  Google Scholar [5] A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, J. Funct. Anal., 18 (1975), 151-176.  doi: 10.1016/0022-1236(75)90022-1.  Google Scholar [6] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [7] C. Guan and F. Yi, A free boundary problem arising from a stochastic optimal control model with bounded dividend rate, Stoch. Anal. Appl., 32 (2014), 742-760.  doi: 10.1080/07362994.2014.922778.  Google Scholar [8] C. Guan and F. Yi, A free boundary problem arising from a stochastic optimal control model under controllable risk, J. Differ. Equ., 260 (2016), 4845-4870.  doi: 10.1016/j.jde.2015.10.040.  Google Scholar [9] B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quant. Financ., 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007.  Google Scholar [10] D. Kelome and A. Swiech, Viscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equation, Appl. Math. Optim., 47 (2003), 253-278.  doi: 10.1007/s00245-003-0764-8.  Google Scholar [11] A. Kolesnichenko and G. Shopina, Valuation of portfolios under uncertain volatility: Black-Scholes-Barenblatt equation and the static hedging, Technical Report, IDE0739, 2007. Google Scholar [12] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar [13] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8.  Google Scholar [14] V. A. Solonnikov, O. A. Ladyzenskaja and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Rusian by Sussian by Smith, S. 1967. Translations of Mathematical Monographs, volume 23, American Mathematical Society, 1968.  Google Scholar [15] M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Math. Meth. of Oper. Res., 51 (2000), 1-42.  doi: 10.1007/s001860050001.  Google Scholar [16] M. Taksar and X. Zhou, Optimal risk and dividend control for a company with a debt liability, Insurance: Mathematics and Economics, 22 (1998), 105-122.  doi: 10.1016/S0167-6687(98)00012-2.  Google Scholar [17] T. Vargiolu, Existence, uniqueness and smoothness for the Black-Scholes-Barenblatt equation, Universita Di Padova, 4 (2001), 315-327.   Google Scholar
Penalty function
Dividend free boundary
Reinsurance free boundary
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