# American Institute of Mathematical Sciences

November  2015, 35(11): 5353-5378. doi: 10.3934/dcds.2015.35.5353

## On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces

 1 School of Management, Fudan University, Shanghai 200433, China

Received  October 2013 Revised  May 2014 Published  May 2015

This paper is concerned with solution in weighted Hölder spaces for backward stochastic partial differential equations (BSPDEs) in a half space. Considering the solution as functional with value in Banach spaces of stochastic processes, and using the methods of partial differential equations (PDEs), we establish the existence and uniqueness of classical solution for BSPDE in functional weighted Hölder spaces.
Citation: Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353
##### References:
 [1] A. Bensoussan, Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions,, Stochastics, 9 (1983), 169.  doi: 10.1080/17442508308833253.  Google Scholar [2] A. Bensoussan, Stochastic Control of Partially Observed Systems,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511526503.  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] N. Dokuchaev, Backward parabolic Itô equations and the second fundamental inequality,, Random Operators and Stochastic Equations, 20 (2012), 69.  doi: 10.1515/rose-2012-0003.  Google Scholar [5] K. Du, J. Qiu and S. Tang, $L^p$ theory for super-parabolic backward stochastic partial differential equations in the whole space,, Applied Mathematics and Optimization, 65 (2012), 175.  doi: 10.1007/s00245-011-9154-9.  Google Scholar [6] K. Du and S. Tang, Strong solution of backward stochastic partial differential equations in $C^2$ domains,, Probability Theory and Related Fields, 154 (2012), 255.  doi: 10.1007/s00440-011-0369-0.  Google Scholar [7] K. Du, S. Tang and Q. Zhang, $W^{m, p}$-solution ($p\geq 2$) of linear degenerate backward stochastic partial differential equations in the whole space,, Journal of Differential Equations, 254 (2013), 2877.  doi: 10.1016/j.jde.2013.01.013.  Google Scholar [8] K. Du and Q. Meng, A revisit to $W_n^2$-theory of super-parabolic backward stochastic partial differential equations in $\mathbbR^d$,, Stochastic Processes and their Applications, 120 (2010), 1996.  doi: 10.1016/j.spa.2010.06.001.  Google Scholar [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Sencond Order,, Springer, (2001).   Google Scholar [10] Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic differential equations,, Probability Theory and Related Fields, 123 (2002), 381.  doi: 10.1007/s004400100193.  Google Scholar [11] Y. Hu and S. Peng, Adapted solution of a backward semi-linear stochastic evolution equations,, Stochastic Analysis and Applications, 9 (1991), 445.  doi: 10.1080/07362999108809250.  Google Scholar [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar [13] J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications,, Stochastic Processes and their Applications, 70 (1997), 59.  doi: 10.1016/S0304-4149(97)00057-4.  Google Scholar [14] J. Ma and J. Yong, On linear, degenerate backward stochastic differential equations,, Probability Theory and Related Fields, 113 (1999), 135.  doi: 10.1007/s004400050205.  Google Scholar [15] R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Hölder classes,, Annals of Probability, 28 (2000), 74.  doi: 10.1214/aop/1019160112.  Google Scholar [16] R. Mikulevicius and H. Pragarauskas, On the Cauchy-Dirichlet problem in half-space for parabolic SPDEs in weighted Hölder spaces,, Stochstic Processes and their Applications, 106 (2003), 185.  doi: 10.1016/S0304-4149(03)00042-5.  Google Scholar [17] S. Peng, Stochastic Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 30 (1992), 284.  doi: 10.1137/0330018.  Google Scholar [18] J. Qiu and S. Tang, Maximum principles for backward stochastic partial differential equations,, Journal of Functional Analysis, 262 (2012), 2436.  doi: 10.1016/j.jfa.2011.12.002.  Google Scholar [19] J. Qiu, S. Tang and Y. You, 2D backward stochastic Navier-Stokes equations with nonlinear forcing,, Stochastic Processes and their Applications, 122 (2012), 334.  doi: 10.1016/j.spa.2011.08.010.  Google Scholar [20] B. Rozovskiĭ, On stochastic partial differential equations,, Sbornik: Mathematics, 25 (1975), 295.   Google Scholar [21] S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations,, SIAM J. Control Optim., 36 (1998), 1596.  doi: 10.1137/S0363012996313100.  Google Scholar [22] S. Tang, A new partially observed stochastic maximum principle,, in 37th IEEE Control and Decision Conference, (1998), 2353.   Google Scholar [23] S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$,, Chinese Annals of Mathematics, 26 (2005), 437.  doi: 10.1142/S025295990500035X.  Google Scholar [24] S. Tang and W. Wei, On the cauchy problem for backward stochastic partial differential equations in Hölder spaces,, to appear in Annals of Probability, ().   Google Scholar [25] X. Zhou, A duality analysis on stochastic partial differential equations,, Journal of Functional Analysis, 103 (1992), 275.  doi: 10.1016/0022-1236(92)90122-Y.  Google Scholar [26] X. Zhou, On the necessary condition 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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##### References:
 [1] A. Bensoussan, Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions,, Stochastics, 9 (1983), 169.  doi: 10.1080/17442508308833253.  Google Scholar [2] A. Bensoussan, Stochastic Control of Partially Observed Systems,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511526503.  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] N. Dokuchaev, Backward parabolic Itô equations and the second fundamental inequality,, Random Operators and Stochastic Equations, 20 (2012), 69.  doi: 10.1515/rose-2012-0003.  Google Scholar [5] K. Du, J. Qiu and S. Tang, $L^p$ theory for super-parabolic backward stochastic partial differential equations in the whole space,, Applied Mathematics and Optimization, 65 (2012), 175.  doi: 10.1007/s00245-011-9154-9.  Google Scholar [6] K. Du and S. Tang, Strong solution of backward stochastic partial differential equations in $C^2$ domains,, Probability Theory and Related Fields, 154 (2012), 255.  doi: 10.1007/s00440-011-0369-0.  Google Scholar [7] K. Du, S. Tang and Q. Zhang, $W^{m, p}$-solution ($p\geq 2$) of linear degenerate backward stochastic partial differential equations in the whole space,, Journal of Differential Equations, 254 (2013), 2877.  doi: 10.1016/j.jde.2013.01.013.  Google Scholar [8] K. Du and Q. Meng, A revisit to $W_n^2$-theory of super-parabolic backward stochastic partial differential equations in $\mathbbR^d$,, Stochastic Processes and their Applications, 120 (2010), 1996.  doi: 10.1016/j.spa.2010.06.001.  Google Scholar [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Sencond Order,, Springer, (2001).   Google Scholar [10] Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic differential equations,, Probability Theory and Related Fields, 123 (2002), 381.  doi: 10.1007/s004400100193.  Google Scholar [11] Y. Hu and S. Peng, Adapted solution of a backward semi-linear stochastic evolution equations,, Stochastic Analysis and Applications, 9 (1991), 445.  doi: 10.1080/07362999108809250.  Google Scholar [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society, (1968).   Google Scholar [13] J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications,, Stochastic Processes and their Applications, 70 (1997), 59.  doi: 10.1016/S0304-4149(97)00057-4.  Google Scholar [14] J. Ma and J. Yong, On linear, degenerate backward stochastic differential equations,, Probability Theory and Related Fields, 113 (1999), 135.  doi: 10.1007/s004400050205.  Google Scholar [15] R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Hölder classes,, Annals of Probability, 28 (2000), 74.  doi: 10.1214/aop/1019160112.  Google Scholar [16] R. Mikulevicius and H. Pragarauskas, On the Cauchy-Dirichlet problem in half-space for parabolic SPDEs in weighted Hölder spaces,, Stochstic Processes and their Applications, 106 (2003), 185.  doi: 10.1016/S0304-4149(03)00042-5.  Google Scholar [17] S. Peng, Stochastic Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 30 (1992), 284.  doi: 10.1137/0330018.  Google Scholar [18] J. Qiu and S. Tang, Maximum principles for backward stochastic partial differential equations,, Journal of Functional Analysis, 262 (2012), 2436.  doi: 10.1016/j.jfa.2011.12.002.  Google Scholar [19] J. Qiu, S. Tang and Y. You, 2D backward stochastic Navier-Stokes equations with nonlinear forcing,, Stochastic Processes and their Applications, 122 (2012), 334.  doi: 10.1016/j.spa.2011.08.010.  Google Scholar [20] B. Rozovskiĭ, On stochastic partial differential equations,, Sbornik: Mathematics, 25 (1975), 295.   Google Scholar [21] S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations,, SIAM J. Control Optim., 36 (1998), 1596.  doi: 10.1137/S0363012996313100.  Google Scholar [22] S. Tang, A new partially observed stochastic maximum principle,, in 37th IEEE Control and Decision Conference, (1998), 2353.   Google Scholar [23] S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$,, Chinese Annals of Mathematics, 26 (2005), 437.  doi: 10.1142/S025295990500035X.  Google Scholar [24] S. Tang and W. Wei, On the cauchy problem for backward stochastic partial differential equations in Hölder spaces,, to appear in Annals of Probability, ().   Google Scholar [25] X. Zhou, A duality analysis on stochastic partial differential equations,, Journal of Functional Analysis, 103 (1992), 275.  doi: 10.1016/0022-1236(92)90122-Y.  Google Scholar [26] X. Zhou, On the necessary condition 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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