Stochastic maximum principle for systems driven by local martingales with spatial parameters

Jian Song; Meng Wang

doi:10.3934/puqr.2021011

Article Contents

2021, Volume 6, Issue 3: 213-236. Doi: 10.3934/puqr.2021011

This issue Previous Article Stein’s method for the law of large numbers under sublinear expectations Next Article An FBSDE approach to market impact games with stochastic parameters

Stochastic maximum principle for systems driven by local martingales with spatial parameters

Jian Song^1,2, , and
Meng Wang^2,

1.
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao 266237, Shandong, China
2.
School of Mathematics, Shandong University, Jinan 250100, Shandong, China

Author Bio: e-mail: wangmeng22@mail.sdu.edu.cn
e-mail: txjsong@sdu.edu.cn
e-mail: txjsong@sdu.edu.cn

Received: June 2021

Accepted: August 25, 2021

Published: September 2021

The authors would like to thank Prof. Mingshang Hu for his helpful discussions. The authors are also grateful to the two anonymous referees for their valuable comments. J. Song is partially supported by Shandong University (Grant No. 11140089963041) and the National Natural Science Foundation of China (Grant No. 12071256).

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Abstract

We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter. Assuming the convexity of the control domain, we obtain the stochastic maximum principle as the necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. The stochastic linear quadratic problem in this setting is also discussed.

Keywords:

Stochastic optimal control,
Stochastic maximum principle,
Local martingale with a spatial parameter.

Mathematics Subject Classification: 93E20, 60H10.

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References

[1]	Peter Bank and Dietmar Baum, Hedging and portfolio optimization in financial markets with a large trader, Math. Finance, 2004, 14(1): 1-18.doi: 10.1111/j.0960-1627.2004.00179.x.
[2]	Jean-Michel Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 1978, 20(1): 62-78.doi: 10.1137/1020004.
[3]	Rainer Buckdahn, Juan Li and Jin Ma, A stochastic maximum principle for general mean-field systems, Appl. Math. Optim., 2016, 74(3): 507-534.doi: 10.1007/s00245-016-9394-9.
[4]	Zengjing Chen and Larry Epstein, Ambiguity, risk, and asset returns in continuous time, Econometrica, 2002, 70(4): 1403-1443.doi: 10.1111/1468-0262.00337.
[5]	Kai Du and Qingxin Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM Journal on Control and Optimization, 2013, 51(6): 4343-4362.doi: 10.1137/120882433.
[6]	N. El Karoui and S.-J. Huang, A general result of existence and uniqueness of backward stochastic differential equations, In: Nicole El Karoui and Laurent Mazliak (ed.), Backward Stochastic Differential Equations, 1997, 364: 27-36, MR1752673.
[7]	Giorgio Fabbri, Fausto Gozzi and Andrzej Swiech, Stochastic Optimal Control in Infinite Dimension, Probability and Stochastic Modelling, Springer, 2017.
[8]	Marco Fuhrman, Ying Hu and Gianmario Tessitore, Stochastic maximum principle for optimal control of SPDEs, Applied Mathematics & Optimization, 2013, 68(2): 181-217.
[9]	Yuecai Han, Shige Peng and Zhen Wu, Maximum principle for backward doubly stochastic control systems with applications, SIAM J. Control Optim., 2010, 48(7): 4224-4241.doi: 10.1137/080743561.
[10]	Mingshang Hu, Stochastic global maximum principle for optimization with recursive utilities, Probab. Uncertain. Quant. Risk, 2017, 2: 1.
[11]	Mingshang Hu, Shaolin Ji and Xiaole Xue, A global stochastic maximum principle for fully coupled forward-backward stochastic systems, SIAM J. Control Optim., 2018, 56(6): 4309-4335.doi: 10.1137/18M1179547.
[12]	Ying Hu and Shige Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 1990, 33(3-4): 159-180.doi: 10.1080/17442509008833671.
[13]	Shaolin Ji and Xun Yu Zhou, A maximum principle for stochastic optimal control with terminal state constraints, and its applications, Commun. Inf. Syst., 2006, 6(4): 321-338.
[14]	H. Kunita, Lectures on Stochastic Flows and Applications, Springer-Verlag, Berlin, 1986, MR867686.
[15]	Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990, MR1070361.
[16]	Juan Li, Stochastic maximum principle in the mean-field controls, Automatica J. IFAC, 2012, 48(2): 366-373.doi: 10.1016/j.automatica.2011.11.006.
[17]	Zongxia Liang, Stochastic differential equations driven by spatial parameters semimartingale with non-Lipschitz local characteristic, Potential Anal., 2007, 26(4): 307-322.doi: 10.1007/s11118-007-9038-4.
[18]	Qi Lv and Xu Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, Springer, 2014.
[19]	Jin Ma and Jiongmin Yong, Forward-backward stochastic differential equations and their applications, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999, 1702: 1-24, MR1704232.
[20]	Anis Matoussi and Michael Scheutzow, Stochastic PDEs driven by nonlinear noise and backward doubly SDEs, J. Theoret. Probab., 2002, 15(1): 1-39.doi: 10.1023/A:1013803104760.
[21]	Thilo Meyer-Brandis, Bernt Ø ksendal and Xun Yu Zhou, A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 2012, 84(5-6): 643-666.doi: 10.1080/17442508.2011.651619.
[22]	Shi Ge Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 1990, 28(4): 966-979.doi: 10.1137/0328054.
[23]	Jian Song, Xiaoming Song and Qi Zhang, Nonlinear Feynman-Kac formulas for stochastic partial differential equations with space-time noise, SIAM Journal on Mathematical Analysis, 2019, 51(2): 955-990.doi: 10.1137/17M1163359.
[24]	Shanjian Tang, The maximum principle for partially observed optimal control of stochastic differential equations, SIAM J. Control Optim., 1998, 36(5): 1596-1617.doi: 10.1137/S0363012996313100.
[25]	Zhen Wu, A general maximum principle for optimal control of forward-backward stochastic systems, Automatica J. IFAC, 2013, 49(5): 1473-1480.doi: 10.1016/j.automatica.2013.02.005.
[26]	Jiongmin Yong and Xun Yu Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Vol. 43, Springer Science & Business Media, 1999.
[27]	Xun Yu Zhou, A unified treatment of maximum principle and dynamic programming in stochastic controls, Stochastics: An International Journal of Probability and Stochastic Processes, 1991, 36(3-4): 137-161.