September  2015, 5(3): 529-555. doi: 10.3934/mcrf.2015.5.529

Transposition method for backward stochastic evolution equations revisited, and its application

1. 

School of Mathematics, Sichuan University, Chengdu 610064, China

2. 

Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

Received  May 2014 Revised  April 2015 Published  July 2015

The main purpose of this paper is to improve our transposition method to solve both vector-valued and operator-valued backward stochastic evolution equations with a general filtration. As its application, we obtain a general Pontryagin-type maximum principle for optimal controls of stochastic evolution equations in infinite dimensions. In particular, we drop the technical assumption appeared in [12, Theorem 9.1]. We also establish a Pontryagin-type maximum principle for a stochastic linear quadratic problems.
Citation: Qi Lü, Xu Zhang. Transposition method for backward stochastic evolution equations revisited, and its application. Mathematical Control & Related Fields, 2015, 5 (3) : 529-555. doi: 10.3934/mcrf.2015.5.529
References:
[1]

A. Al-Hussein, Backward stochastic partial differential equations driven by infinite dimensional martingales and applications,, Stochastics, 81 (2009), 601. doi: 10.1080/17442500903370202. Google Scholar

[2]

A. Bensoussan, Stochastic maximum principle for distributed parameter systems,, J. Franklin Inst., 315 (1983), 387. doi: 10.1016/0016-0032(83)90059-5. Google Scholar

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[4]

K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations,, SIAM J. Control Optim., 51 (2013), 4343. doi: 10.1137/120882433. Google Scholar

[5]

M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs,, Appl. Math. Optim., 68 (2013), 181. doi: 10.1007/s00245-013-9203-7. Google Scholar

[6]

Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems,, Stochastics Stochastics Rep., 33 (1990), 159. doi: 10.1080/17442509008833671. Google Scholar

[7]

Y. Hu and S. Peng, Adapted solution of backward semilinear stochastic evolution equations,, Stoch. Anal. Appl., 9 (1991), 445. doi: 10.1080/07362999108809250. Google Scholar

[8]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1,, Recherches en Mathématiques Appliquées, (1988). Google Scholar

[9]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I,, Die Grundlehren der mathematischen Wissenschaften, 181 (1972). Google Scholar

[10]

Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals,, J. Eur. Math. Soc, 14 (2012), 1795. doi: 10.4171/JEMS/347. Google Scholar

[11]

Q. Lü and X. Zhang, Well-posedness of backward stochastic differential equations with general filtration,, J. Differential Equations, 254 (2013), 3200. doi: 10.1016/j.jde.2013.01.010. Google Scholar

[12]

Q. Lü and X. Zhang, General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions,, Springer Briefs in Mathematics, (2014). doi: 10.1007/978-3-319-06632-5. Google Scholar

[13]

J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications,, Lecture Notes in Math. vol. 1702. Springer-Verlag, 1702 (1999). Google Scholar

[14]

N. I. Mahmudova and M. A. McKibben, On backward stochastic evolution equations in Hilbert spaces and optimal control,, Nonlinear Anal., 67 (2007), 1260. doi: 10.1016/j.na.2006.07.013. Google Scholar

[15]

S. Peng, A general stochastic maximum principle for optimal control problems,, SIAM J. Control Optim., 28 (1990), 966. doi: 10.1137/0328054. Google Scholar

[16]

S. Tang and X. Li, Maximum principle for optimal control of distributed parameter stochastic systems with random jumps,, in Differential Equations, 152 (1994), 867. Google Scholar

[17]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar

[18]

X. Y. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations,, SIAM J. Control Optim., 31 (1993), 1462. doi: 10.1137/0331068. Google Scholar

show all references

References:
[1]

A. Al-Hussein, Backward stochastic partial differential equations driven by infinite dimensional martingales and applications,, Stochastics, 81 (2009), 601. doi: 10.1080/17442500903370202. Google Scholar

[2]

A. Bensoussan, Stochastic maximum principle for distributed parameter systems,, J. Franklin Inst., 315 (1983), 387. doi: 10.1016/0016-0032(83)90059-5. Google Scholar

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[4]

K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations,, SIAM J. Control Optim., 51 (2013), 4343. doi: 10.1137/120882433. Google Scholar

[5]

M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs,, Appl. Math. Optim., 68 (2013), 181. doi: 10.1007/s00245-013-9203-7. Google Scholar

[6]

Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems,, Stochastics Stochastics Rep., 33 (1990), 159. doi: 10.1080/17442509008833671. Google Scholar

[7]

Y. Hu and S. Peng, Adapted solution of backward semilinear stochastic evolution equations,, Stoch. Anal. Appl., 9 (1991), 445. doi: 10.1080/07362999108809250. Google Scholar

[8]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1,, Recherches en Mathématiques Appliquées, (1988). Google Scholar

[9]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I,, Die Grundlehren der mathematischen Wissenschaften, 181 (1972). Google Scholar

[10]

Q. Lü, J. Yong and X. Zhang, Representation of Itô integrals by Lebesgue/Bochner integrals,, J. Eur. Math. Soc, 14 (2012), 1795. doi: 10.4171/JEMS/347. Google Scholar

[11]

Q. Lü and X. Zhang, Well-posedness of backward stochastic differential equations with general filtration,, J. Differential Equations, 254 (2013), 3200. doi: 10.1016/j.jde.2013.01.010. Google Scholar

[12]

Q. Lü and X. Zhang, General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions,, Springer Briefs in Mathematics, (2014). doi: 10.1007/978-3-319-06632-5. Google Scholar

[13]

J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications,, Lecture Notes in Math. vol. 1702. Springer-Verlag, 1702 (1999). Google Scholar

[14]

N. I. Mahmudova and M. A. McKibben, On backward stochastic evolution equations in Hilbert spaces and optimal control,, Nonlinear Anal., 67 (2007), 1260. doi: 10.1016/j.na.2006.07.013. Google Scholar

[15]

S. Peng, A general stochastic maximum principle for optimal control problems,, SIAM J. Control Optim., 28 (1990), 966. doi: 10.1137/0328054. Google Scholar

[16]

S. Tang and X. Li, Maximum principle for optimal control of distributed parameter stochastic systems with random jumps,, in Differential Equations, 152 (1994), 867. Google Scholar

[17]

J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Springer-Verlag, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar

[18]

X. Y. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations,, SIAM J. Control Optim., 31 (1993), 1462. doi: 10.1137/0331068. Google Scholar

[1]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[2]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021

[3]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004

[4]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613

[5]

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

[6]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067

[7]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[8]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195

[9]

Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043

[10]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499

[11]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018

[12]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174

[13]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[14]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[15]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565

[16]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320

[17]

Goro Akagi, Jun Kobayashi, Mitsuharu Ôtani. Principle of symmetric criticality and evolution equations. Conference Publications, 2003, 2003 (Special) : 1-10. doi: 10.3934/proc.2003.2003.1

[18]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77

[19]

Jiongmin Yong. Forward-backward evolution equations and applications. Mathematical Control & Related Fields, 2016, 6 (4) : 653-704. doi: 10.3934/mcrf.2016019

[20]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]