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A stochastic maximum principle for linear quadratic problem with nonconvex control domain
Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong, 250100, China |
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.
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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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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