# 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:

show all references

##### References:
Penalty function
Dividend free boundary
Reinsurance free boundary
 [1] Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations & Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017 [2] Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019080 [3] T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675 [4] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [5] Martin Brokate, Pavel Krejčí. Optimal control of ODE systems involving a rate independent variational inequality. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 331-348. doi: 10.3934/dcdsb.2013.18.331 [6] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 [7] Yan Zhang, Yonghong Wu, Benchawan Wiwatanapataphee, Francisca Angkola. Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-31. doi: 10.3934/jimo.2018141 [8] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [9] Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122 [10] Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605 [11] Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 [12] Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 [13] Jingzhen Liu, Ka-Fai Cedric Yiu, Tak Kuen Siu, Wai-Ki Ching. Optimal insurance in a changing economy. Mathematical Control & Related Fields, 2014, 4 (2) : 187-202. doi: 10.3934/mcrf.2014.4.187 [14] Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71 [15] Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645 [16] Chonghu Guan, Xun Li, Zuo Quan Xu, Fahuai Yi. A stochastic control problem and related free boundaries in finance. Mathematical Control & Related Fields, 2017, 7 (4) : 563-584. doi: 10.3934/mcrf.2017021 [17] William G. Litvinov. Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions. Journal of Industrial & Management Optimization, 2011, 7 (2) : 291-315. doi: 10.3934/jimo.2011.7.291 [18] Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133 [19] Stanisław Migórski. A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications, 2013, 2013 (special) : 545-554. doi: 10.3934/proc.2013.2013.545 [20] Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320

2018 Impact Factor: 1.292

## Metrics

• HTML views (292)
• Cited by (0)

• on AIMS