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

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