# 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 and 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. [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. [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. [4] A. Friedman, Partial Differential Equaions of Parabolic Type, Prentice-Hall Inc., 1964. [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. [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. [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. [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. [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. [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. [11] A. Kolesnichenko and G. Shopina, Valuation of portfolios under uncertain volatility: Black-Scholes-Barenblatt equation and the static hedging, Technical Report, IDE0739, 2007. [12] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [13] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8. [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. [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. [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. [17] T. Vargiolu, Existence, uniqueness and smoothness for the Black-Scholes-Barenblatt equation, Universita Di Padova, 4 (2001), 315-327.

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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. [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. [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. [4] A. Friedman, Partial Differential Equaions of Parabolic Type, Prentice-Hall Inc., 1964. [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. [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. [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. [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. [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. [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. [11] A. Kolesnichenko and G. Shopina, Valuation of portfolios under uncertain volatility: Black-Scholes-Barenblatt equation and the static hedging, Technical Report, IDE0739, 2007. [12] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [13] H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89500-8. [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. [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. [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. [17] T. Vargiolu, Existence, uniqueness and smoothness for the Black-Scholes-Barenblatt equation, Universita Di Padova, 4 (2001), 315-327.
Penalty function
Dividend free boundary
Reinsurance free boundary
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