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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. BieleckiH. JinS. 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. DaiZ. 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. GuanX. LiZ. 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. GuanF. 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. HuJ. 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. LiX. 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. YangF. 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. BieleckiH. JinS. 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. DaiZ. 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. GuanX. LiZ. 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. GuanF. 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. HuJ. 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. LiX. 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. YangF. 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.

Figure 1.  Free boundaries with various $ k $ and $ b $
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