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

Shaolin Ji; Xiaole Xue

doi:10.3934/mcrf.2019022

Article Contents

2019, Volume 9, Issue 3: 495-507. Doi: 10.3934/mcrf.2019022

This issue Previous Article Controllability for a string with attached masses and Riesz bases for asymmetric spaces Next Article Determining the shape of a solid of revolution

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
^* Corresponding author: Xiaole Xue

Received: May 31, 2017

Revised: May 31, 2018

Published: September 2019

Research supported by NSF (Nos. 11571203, 11871458).

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Full Text(HTML)

Related Papers

Cited by

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.