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: |
[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