American Institute of Mathematical Sciences

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

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

Shaolin Ji and  Xiaole Xue ,

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.

Keywords: Stochastic maximum principle, stochastic linear quadratic problem, convex perturbation, backward stochastic differential equation.
Mathematics Subject Classification: Primary: 60H10, 93E20; Secondary: 49N15.
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. Bielecki, H. Jin, S. 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. Chen, X. 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 Karoui, S. 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. 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]

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. Bielecki, H. Jin, S. 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. Chen, X. 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 Karoui, S. 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. 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]

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.

[1]

Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028
[2]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[3]

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
[4]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035
[5]

Renzhi Qiu, Shanjian Tang. The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 3-. doi: 10.1186/s41546-019-0037-3
[6]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control and Related Fields, 2022, 12 (2) : 475-493. doi: 10.3934/mcrf.2021031
[7]

Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7
[8]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[9]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100
[10]

Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial and Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27
[11]

Weijun Meng, Jingtao Shi. A linear quadratic stochastic Stackelberg differential game with time delay. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021035
[12]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[13]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581
[14]

Jian Song, Meng Wang. Stochastic maximum principle for systems driven by local martingales with spatial parameters. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 213-236. doi: 10.3934/puqr.2021011
[15]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control and Related Fields, 2021, 11 (4) : 829-855. doi: 10.3934/mcrf.2020048
[16]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
[17]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[18]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[19]

Joscha Diehl, Jianfeng Zhang. Backward stochastic differential equations with Young drift. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 5-. doi: 10.1186/s41546-017-0016-5
[20]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (360)
  • HTML views (589)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy