September  2021, 6(3): 213-236. doi: 10.3934/puqr.2021011

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

1. 

Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao 266237, Shandong, China

2. 

School of Mathematics, Shandong University, Jinan 250100, Shandong, China

e-mail: txjsong@sdu.edu.cn

Received  June 02, 2021 Accepted  August 26, 2021 Published  September 2021

Fund Project: 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).

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.

Citation: 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
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.  Google Scholar

[2]

Jean-Michel Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 1978, 20(1): 62-78. doi: 10.1137/1020004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

Giorgio Fabbri, Fausto Gozzi and Andrzej Swiech, Stochastic Optimal Control in Infinite Dimension, Probability and Stochastic Modelling, Springer, 2017. Google Scholar

[8]

Marco Fuhrman, Ying Hu and Gianmario Tessitore, Stochastic maximum principle for optimal control of SPDEs, Applied Mathematics & Optimization, 2013, 68(2): 181-217. Google Scholar

[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.  Google Scholar

[10]

Mingshang Hu, Stochastic global maximum principle for optimization with recursive utilities, Probab. Uncertain. Quant. Risk, 2017, 2: 1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

H. Kunita, Lectures on Stochastic Flows and Applications, Springer-Verlag, Berlin, 1986, MR867686. Google Scholar

[15]

Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990, MR1070361. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Qi Lv and Xu Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, Springer, 2014. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Jiongmin Yong and Xun Yu Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Vol. 43, Springer Science & Business Media, 1999. Google Scholar

[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. Google Scholar

show all references

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.  Google Scholar

[2]

Jean-Michel Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 1978, 20(1): 62-78. doi: 10.1137/1020004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

Giorgio Fabbri, Fausto Gozzi and Andrzej Swiech, Stochastic Optimal Control in Infinite Dimension, Probability and Stochastic Modelling, Springer, 2017. Google Scholar

[8]

Marco Fuhrman, Ying Hu and Gianmario Tessitore, Stochastic maximum principle for optimal control of SPDEs, Applied Mathematics & Optimization, 2013, 68(2): 181-217. Google Scholar

[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.  Google Scholar

[10]

Mingshang Hu, Stochastic global maximum principle for optimization with recursive utilities, Probab. Uncertain. Quant. Risk, 2017, 2: 1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

H. Kunita, Lectures on Stochastic Flows and Applications, Springer-Verlag, Berlin, 1986, MR867686. Google Scholar

[15]

Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990, MR1070361. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Qi Lv and Xu Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, Springer, 2014. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Jiongmin Yong and Xun Yu Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Vol. 43, Springer Science & Business Media, 1999. Google Scholar

[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. Google Scholar

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