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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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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