# American Institute of Mathematical Sciences

• Previous Article
Multi-aircraft cooperative path planning for maneuvering target detection
• JIMO Home
• This Issue
• Next Article
Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction
May  2022, 18(3): 1915-1934. doi: 10.3934/jimo.2021049

## Free boundary problem for an optimal investment problem with a borrowing constraint

 1 School of Mathematics, Jiaying University, Meizhou 514015, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

* Corresponding author: Rui Zhou

Received  June 2020 Revised  December 2020 Published  May 2022 Early access  March 2021

Fund Project: The work is supported by NNSF of China (No.11901244 and No.11901093), Universities and Colleges Special Innovation Project of Guangdong Province (No.2019KTSCX166), and Research Grants Council of Hong kong under grant 15213218 and 15215319

This paper considers an optimal investment problem under CRRA utility with a borrowing constraint. We formulate it into a free boundary problem consisting of a fully nonlinear equation and a linear equation. We prove the existence and uniqueness of the classical solution and present the condition for the existence of the free boundary under a linear constraint on a borrowing rate. Furthermore, we prove that the free boundary is continuous and smooth when the relative risk aversion coefficient is sufficiently small.

Citation: Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1915-1934. doi: 10.3934/jimo.2021049
##### References:
 [1] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0. [2] T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x. [3] M. Dai, Z. Q. Xu and X. Y. Zhou, Continuous-time Markowitz's model with transaction costs, SIAM Journal on Financial Mathematics, 1 (2010), 96-125.  doi: 10.1137/080742889. [4] M. Dai and F. Yi, Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem, Journal of Differential Equations, 246 (2009), 1445-1469.  doi: 10.1016/j.jde.2008.11.003. [5] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [6] A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, Journal of Functional Analysis, 18 (1975), 151-176.  doi: 10.1016/0022-1236(75)90022-1. [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^nd$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [8] C. Guan, On a free boundary problem for an optimal investment problem with different interest rates, Communications in Mathematical Sciences, 18 (2020), 31-54.  doi: 10.4310/CMS.2020.v18.n1.a2. [9] C. Guan, X. Li, Z. Q. Xu and F. Yi, A stochastic control problem and related free boundaries in finance, Mathematical Control & Related Fields, 7 (2017), 563-584.  doi: 10.3934/mcrf.2017021. [10] C. Guan, F. Yi and J. Chen, Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain, Journal of Differential Equations, 266 (2019), 1245-1284.  doi: 10.1016/j.jde.2018.07.070. [11] B. Hu, J. Liang and Y. Wu, A free boundary problem for corporate bond with credit rating migration, Journal of Mathematical Analysis and Applications, 428 (2015), 896-909.  doi: 10.1016/j.jmaa.2015.03.040. [12] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968. doi: 10.1090/mmono/023. [13] X. Li and Z. Q. Xu, Continuous-time Markowitz's model with constraints on wealth and portfolio, Operations Research Letters, 44 (2016), 729-736.  doi: 10.1016/j.orl.2016.09.004. [14] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504. [15] G. M. Lieberman, Second Order Parabolic Differential Equations, World scientific, 1996. doi: 10.1142/3302. [16] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [17] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [18] A. O. Olejnik and E. V. Radkevic, Second Order Equations with Nonnegative Characteristic Form, AMS, New York-London, 1973. doi: 10.1007/978-1-4684-8965-1. [19] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51 (1969), 239-246.  doi: 10.2307/1926559. [20] M. I. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.  doi: 10.1007/s001860050001. [21] Z. Yang, F. Yi and M. Dai, A parabolic variational inequality arising from the valuation of strike reset options, Journal of Differential Equations, 230 (2006), 481-501.  doi: 10.1016/j.jde.2006.07.026. [22] T. Zariphopoulou, Consumption-investment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 59-85.  doi: 10.1137/S0363012991218827.

show all references

##### References:
 [1] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0. [2] T. R. Bielecki, H. Jin, S. R. Pliska and X. Y. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x. [3] M. Dai, Z. Q. Xu and X. Y. Zhou, Continuous-time Markowitz's model with transaction costs, SIAM Journal on Financial Mathematics, 1 (2010), 96-125.  doi: 10.1137/080742889. [4] M. Dai and F. Yi, Finite-horizon optimal investment with transaction costs: A parabolic double obstacle problem, Journal of Differential Equations, 246 (2009), 1445-1469.  doi: 10.1016/j.jde.2008.11.003. [5] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [6] A. Friedman, Parabolic variational inequalities in one space dimension and smoothness of the free boundary, Journal of Functional Analysis, 18 (1975), 151-176.  doi: 10.1016/0022-1236(75)90022-1. [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^nd$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [8] C. Guan, On a free boundary problem for an optimal investment problem with different interest rates, Communications in Mathematical Sciences, 18 (2020), 31-54.  doi: 10.4310/CMS.2020.v18.n1.a2. [9] C. Guan, X. Li, Z. Q. Xu and F. Yi, A stochastic control problem and related free boundaries in finance, Mathematical Control & Related Fields, 7 (2017), 563-584.  doi: 10.3934/mcrf.2017021. [10] C. Guan, F. Yi and J. Chen, Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain, Journal of Differential Equations, 266 (2019), 1245-1284.  doi: 10.1016/j.jde.2018.07.070. [11] B. Hu, J. Liang and Y. Wu, A free boundary problem for corporate bond with credit rating migration, Journal of Mathematical Analysis and Applications, 428 (2015), 896-909.  doi: 10.1016/j.jmaa.2015.03.040. [12] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968. doi: 10.1090/mmono/023. [13] X. Li and Z. Q. Xu, Continuous-time Markowitz's model with constraints on wealth and portfolio, Operations Research Letters, 44 (2016), 729-736.  doi: 10.1016/j.orl.2016.09.004. [14] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.  doi: 10.1137/S0363012900378504. [15] G. M. Lieberman, Second Order Parabolic Differential Equations, World scientific, 1996. doi: 10.1142/3302. [16] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [17] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [18] A. O. Olejnik and E. V. Radkevic, Second Order Equations with Nonnegative Characteristic Form, AMS, New York-London, 1973. doi: 10.1007/978-1-4684-8965-1. [19] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51 (1969), 239-246.  doi: 10.2307/1926559. [20] M. I. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.  doi: 10.1007/s001860050001. [21] Z. Yang, F. Yi and M. Dai, A parabolic variational inequality arising from the valuation of strike reset options, Journal of Differential Equations, 230 (2006), 481-501.  doi: 10.1016/j.jde.2006.07.026. [22] T. Zariphopoulou, Consumption-investment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 59-85.  doi: 10.1137/S0363012991218827.
Free boundaries with various $k$ and $b$
 [1] Jingzhen Liu, Ka-Fai Cedric Yiu, Kok Lay Teo. Optimal investment-consumption problem with constraint. Journal of Industrial and Management Optimization, 2013, 9 (4) : 743-768. doi: 10.3934/jimo.2013.9.743 [2] Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 [3] Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial and Management Optimization, 2022, 18 (2) : 933-967. doi: 10.3934/jimo.2021003 [4] Jingzhen Liu, Shiqi Yan, Shan Jiang, Jiaqin Wei. Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022040 [5] Jingzhen Liu, Lihua Bai, Ka-Fai Cedric Yiu. Optimal investment with a value-at-risk constraint. Journal of Industrial and Management Optimization, 2012, 8 (3) : 531-547. doi: 10.3934/jimo.2012.8.531 [6] Zuo Quan Xu, Fahuai Yi. An optimal consumption-investment model with constraint on consumption. Mathematical Control and Related Fields, 2016, 6 (3) : 517-534. doi: 10.3934/mcrf.2016014 [7] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 [8] Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial and Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044 [9] Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial and Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067 [10] 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 [11] Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080 [12] Zhen-Zhen Tao, Bing Sun. Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021220 [13] Zhen-Zhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022080 [14] Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations and Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017 [15] 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 [16] Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks and Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605 [17] Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control and Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185 [18] Xiaoshan Chen, Xun Li, Fahuai Yi. Optimal stopping investment with non-smooth utility over an infinite time horizon. Journal of Industrial and Management Optimization, 2019, 15 (1) : 81-96. doi: 10.3934/jimo.2018033 [19] Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2139-2159. doi: 10.3934/jimo.2020062 [20] Shuaiqi Zhang, Jie Xiong, Xin Zhang. Optimal investment problem with delay under partial information. Mathematical Control and Related Fields, 2020, 10 (2) : 365-378. doi: 10.3934/mcrf.2020001

2020 Impact Factor: 1.801