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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  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 & Management Optimization, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] G. M. Lieberman, Second Order Parabolic Differential Equations, World scientific, 1996. doi: 10.1142/3302.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] T. Zariphopoulou, Consumption-investment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 59-85.  doi: 10.1137/S0363012991218827.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] G. M. Lieberman, Second Order Parabolic Differential Equations, World scientific, 1996. doi: 10.1142/3302.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] T. Zariphopoulou, Consumption-investment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 59-85.  doi: 10.1137/S0363012991218827.  Google Scholar
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