# American Institute of Mathematical Sciences

September & December  2018, 8(3&4): 739-751. doi: 10.3934/mcrf.2018032

## Nonlinear backward stochastic evolutionary equations driven by a space-time white noise

 1 Institut de Recherche Mathématique de Rennes, Université Rennes 1, 35042 Rennes Cedex, France 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China 3 Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

* Corresponding authorr: Shanjian Tang

Received  August 2017 Revised  April 2018 Published  September 2018

Fund Project: Ying Hu's research is partially supported by Lebesgue Center of Mathematics "Investissements d'avenir" Program (No. ANR-11-LABX-0020-01), by ANR CAESARS (No. ANR-15-CE05-0024) and by ANR MFG (No. ANR-16-CE40-0015-01). Shanjian Tang's research is partially supported by National Science Foundation of China (No. 11631004) and Science and Technology Commission of Shanghai Municipality (No. 14XD1400400).

We study the well solvability of nonlinear backward stochastic evolutionary equations driven by a space-time white noise. We first establish a novel a priori estimate for solution of linear backward stochastic evolutionary equations, and then give an existence and uniqueness result for nonlinear backward stochastic evolutionary equations. A dual argument plays a crucial role in the proof of these results. Finally, an example is given to illustrate the existence and uniqueness result.

Citation: Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control and Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032
##### References:
 [1] A. Bensoussan, Stochastic maximum principle for distributed parameter systems, J. Franklin Inst., 315 (1983), 387-406.  doi: 10.1016/0016-0032(83)90059-5. [2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. [3] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. [4] K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362.  doi: 10.1137/120882433. [5] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217.  doi: 10.1007/s00245-013-9203-7. [6] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of partial differential equations driven by white noise, Stoch. Partial Differ. Equ. Anal. Comput., 6 (2018), 255-285.  doi: 10.1007/s40072-017-0108-3. [7] G. Guatteri, Stochastic maximum principle for SPDEs with noise and control on the boundary, Systems Control Lett., 60 (2011), 198-204.  doi: 10.1016/j.sysconle.2011.01.001. [8] Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 33 (1990), 159-180.  doi: 10.1080/17442509008833671. [9] Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Anal. Appl., 9 (1991), 445-459.  doi: 10.1080/07362999108809250. [10] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band, 181. Springer-Verlag, New York, 1972. [11] Q. Lü and X. Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, SpringerBriefs in Mathematices. Springer, Cham, 2014. doi: 10.1007/978-3-319-06632-5. [12] Q. Lü and X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Math. Control Relat. Fields, 5 (2015), 529-555.  doi: 10.3934/mcrf.2015.5.529. [13] S. Tang and X. Li, Maximum principle for optimal control of distributed parameter stochastic systems with random jumps, Differential Equations, Dynamical Systems, and Control Science, 867–890, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, 1994. [14] X. Y. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1993), 1462-1478.  doi: 10.1137/0331068.

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##### References:
 [1] A. Bensoussan, Stochastic maximum principle for distributed parameter systems, J. Franklin Inst., 315 (1983), 387-406.  doi: 10.1016/0016-0032(83)90059-5. [2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. [3] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829. [4] K. Du and Q. Meng, A maximum principle for optimal control of stochastic evolution equations, SIAM J. Control Optim., 51 (2013), 4343-4362.  doi: 10.1137/120882433. [5] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of SPDEs, Appl. Math. Optim., 68 (2013), 181-217.  doi: 10.1007/s00245-013-9203-7. [6] M. Fuhrman, Y. Hu and G. Tessitore, Stochastic maximum principle for optimal control of partial differential equations driven by white noise, Stoch. Partial Differ. Equ. Anal. Comput., 6 (2018), 255-285.  doi: 10.1007/s40072-017-0108-3. [7] G. Guatteri, Stochastic maximum principle for SPDEs with noise and control on the boundary, Systems Control Lett., 60 (2011), 198-204.  doi: 10.1016/j.sysconle.2011.01.001. [8] Y. Hu and S. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 33 (1990), 159-180.  doi: 10.1080/17442509008833671. [9] Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Anal. Appl., 9 (1991), 445-459.  doi: 10.1080/07362999108809250. [10] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band, 181. Springer-Verlag, New York, 1972. [11] Q. Lü and X. Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, SpringerBriefs in Mathematices. Springer, Cham, 2014. doi: 10.1007/978-3-319-06632-5. [12] Q. Lü and X. Zhang, Transposition method for backward stochastic evolution equations revisited, and its application, Math. Control Relat. Fields, 5 (2015), 529-555.  doi: 10.3934/mcrf.2015.5.529. [13] S. Tang and X. Li, Maximum principle for optimal control of distributed parameter stochastic systems with random jumps, Differential Equations, Dynamical Systems, and Control Science, 867–890, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, 1994. [14] X. Y. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1993), 1462-1478.  doi: 10.1137/0331068.
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