September  2019, 9(3): 495-507. doi: 10.3934/mcrf.2019022

A stochastic maximum principle for linear quadratic problem with nonconvex control domain

Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong, 250100, China

* Corresponding author: Xiaole Xue

Received  June 2017 Revised  June 2018 Published  April 2019

Fund Project: Research supported by NSF (Nos. 11571203, 11871458).

This paper considers the stochastic linear quadratic optimal control problem in which the control domain is nonconvex. By the functional analysis and convex perturbation methods, we establish a novel maximum principle. The application of the proposed maximum principle is illustrated through a work-out example.

Citation: Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control and Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
References:
[1]

A. Bensoussan, Lectures on stochastic control, Nonlinear filtering and stochastic control, Springer Berlin Heidelberg, 972 (1982), 1-62. 

[2]

A. Bensoussan, Stochastic maximum principle for distributed parameter systems, Journal of the Franklin Institute, 315 (1983), 387-406.  doi: 10.1016/0016-0032(83)90059-5.

[3]

T. R. BieleckiH. JinS. R. Pliska and X. Y. Zhou, Continuous time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x.

[4]

J. M. Bismut, Théorie probabiliste du contrôle des diffusions, Mem. Amer. Math. Soc., 167, (1976). doi: 10.1090/memo/0167.

[5]

S. ChenX. Li and X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, SIAM Journal on Control and Optimization, 36 (1998), 1685-1702.  doi: 10.1137/S0363012996310478.

[6]

D. Duffie and M. O. Jackson, Optimal hedging and equilibrium in a dynamic futures market, Journal of Economic Dynamics and Control, 14 (1990), 21-33.  doi: 10.1016/0165-1889(90)90003-Y.

[7]

D. Duffie and H. R. Richardson, Mean-variance hedging in continuous time, The Annals of Applied Probability, 1 (1991), 1-15.  doi: 10.1214/aoap/1177005978.

[8]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[9]

E. J. Elton and M. J. Gruber, Finance as a Dynamic Process, Englewood Cliffs, NJ: Prentice-Hall, 1975.

[10]

H. Föllmer and D. Sondermann, Hedging of non-redundant contingent claims, Contributions to Mathematical Economics, North Holland. Hildebrandt and Mas-Colell, 1986,205–223.

[11]

R. R. Grauer and N. H. Hakansson, On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: A comparison of returns and investment policies, Management Science, 39 (1993), 856-871. 

[12]

A. J. Heunis, Quadratic minimization with portfolio and terminal wealth constraints, Annals of Finance, 11 (2015), 243-282.  doi: 10.1007/s10436-014-0254-9.

[13]

S. Ji and X. Y. Zhou, A maximum principle for stochastic optimal control with terminal state constraints and its applications, Communications in Information Systems, 6 (2006), 321-337.  doi: 10.4310/CIS.2006.v6.n4.a4.

[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]

X. Li and Z. Q. Xu, Continuous-time mean-variance portfolio selection with constraints on wealth and portfolio, Oper. Res. Lett., 44 (2016), 729-736.  doi: 10.1016/j.orl.2016.09.004.

[16]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. 

[17]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Cowles Foundation for Research in Economics at Yale University, Monograph 16 John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1959.

[18]

M. J. Optimal, multiperiod portfolio policies, The Journal of Business, 41 (1968), 215-229. 

[19]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM Journal on Control and Optimization, 28 (1990), 966-979.  doi: 10.1137/0328054.

[20]

S. Peng, Backward stochastic differential equations and applications to optimal control, Applied Mathematics and Optimization, 27 (1993), 125-144.  doi: 10.1007/BF01195978.

[21]

P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 1969,239–246.

[22]

W. M. Wonham, On the separation theorem of stochastic control, SIAM Journal on Control, 6 (1968), 312-326.  doi: 10.1137/0306023.

[23]

Z. Wu, A general maximum principle for optimal control of forward-backward stochastic systems, Automatica, 49 (2013), 1473-1480.  doi: 10.1016/j.automatica.2013.02.005.

[24]

J. Yong, Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions, SIAM Journal on Control and Optimization, 48 (2010), 4119-4156.  doi: 10.1137/090763287.

[25]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamilton Systems and HJB Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[26]

X. Y. Zhou and D. Li, Explicit efficient frontier of a continuous-time mean-variance portfolio selection problem, in Control of Distributed Parameter and Stochastic Systems, Springer US, 1999,323–330.

show all references

References:
[1]

A. Bensoussan, Lectures on stochastic control, Nonlinear filtering and stochastic control, Springer Berlin Heidelberg, 972 (1982), 1-62. 

[2]

A. Bensoussan, Stochastic maximum principle for distributed parameter systems, Journal of the Franklin Institute, 315 (1983), 387-406.  doi: 10.1016/0016-0032(83)90059-5.

[3]

T. R. BieleckiH. JinS. R. Pliska and X. Y. Zhou, Continuous time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.  doi: 10.1111/j.0960-1627.2005.00218.x.

[4]

J. M. Bismut, Théorie probabiliste du contrôle des diffusions, Mem. Amer. Math. Soc., 167, (1976). doi: 10.1090/memo/0167.

[5]

S. ChenX. Li and X. Y. Zhou, Stochastic linear quadratic regulators with indefinite control weight costs, SIAM Journal on Control and Optimization, 36 (1998), 1685-1702.  doi: 10.1137/S0363012996310478.

[6]

D. Duffie and M. O. Jackson, Optimal hedging and equilibrium in a dynamic futures market, Journal of Economic Dynamics and Control, 14 (1990), 21-33.  doi: 10.1016/0165-1889(90)90003-Y.

[7]

D. Duffie and H. R. Richardson, Mean-variance hedging in continuous time, The Annals of Applied Probability, 1 (1991), 1-15.  doi: 10.1214/aoap/1177005978.

[8]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[9]

E. J. Elton and M. J. Gruber, Finance as a Dynamic Process, Englewood Cliffs, NJ: Prentice-Hall, 1975.

[10]

H. Föllmer and D. Sondermann, Hedging of non-redundant contingent claims, Contributions to Mathematical Economics, North Holland. Hildebrandt and Mas-Colell, 1986,205–223.

[11]

R. R. Grauer and N. H. Hakansson, On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: A comparison of returns and investment policies, Management Science, 39 (1993), 856-871. 

[12]

A. J. Heunis, Quadratic minimization with portfolio and terminal wealth constraints, Annals of Finance, 11 (2015), 243-282.  doi: 10.1007/s10436-014-0254-9.

[13]

S. Ji and X. Y. Zhou, A maximum principle for stochastic optimal control with terminal state constraints and its applications, Communications in Information Systems, 6 (2006), 321-337.  doi: 10.4310/CIS.2006.v6.n4.a4.

[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]

X. Li and Z. Q. Xu, Continuous-time mean-variance portfolio selection with constraints on wealth and portfolio, Oper. Res. Lett., 44 (2016), 729-736.  doi: 10.1016/j.orl.2016.09.004.

[16]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. 

[17]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Cowles Foundation for Research in Economics at Yale University, Monograph 16 John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1959.

[18]

M. J. Optimal, multiperiod portfolio policies, The Journal of Business, 41 (1968), 215-229. 

[19]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM Journal on Control and Optimization, 28 (1990), 966-979.  doi: 10.1137/0328054.

[20]

S. Peng, Backward stochastic differential equations and applications to optimal control, Applied Mathematics and Optimization, 27 (1993), 125-144.  doi: 10.1007/BF01195978.

[21]

P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 1969,239–246.

[22]

W. M. Wonham, On the separation theorem of stochastic control, SIAM Journal on Control, 6 (1968), 312-326.  doi: 10.1137/0306023.

[23]

Z. Wu, A general maximum principle for optimal control of forward-backward stochastic systems, Automatica, 49 (2013), 1473-1480.  doi: 10.1016/j.automatica.2013.02.005.

[24]

J. Yong, Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions, SIAM Journal on Control and Optimization, 48 (2010), 4119-4156.  doi: 10.1137/090763287.

[25]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamilton Systems and HJB Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.

[26]

X. Y. Zhou and D. Li, Explicit efficient frontier of a continuous-time mean-variance portfolio selection problem, in Control of Distributed Parameter and Stochastic Systems, Springer US, 1999,323–330.

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