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

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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

Wenning Wei ^1,

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

Received October 2013 Revised May 2014 Published May 2015

Abstract
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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.

Keywords: Cauchy-Dirichlet problem, heat potential., weighted Hölder space, Backward stochastic partial differential equations, backward stochastic differential equations.

Mathematics Subject Classification: Primary: 60H15; Secondary: 35R6.

Citation: Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete and Continuous Dynamical Systems, 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), 169-222. doi: 10.1080/17442508308833253.
[2]	A. Bensoussan, Stochastic Control of Partially Observed Systems, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511526503.
[3]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.
[4]	N. Dokuchaev, Backward parabolic Itô equations and the second fundamental inequality, Random Operators and Stochastic Equations, 20 (2012), 69-102. doi: 10.1515/rose-2012-0003.
[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), 175-219. doi: 10.1007/s00245-011-9154-9.
[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), 255-285. doi: 10.1007/s00440-011-0369-0.
[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), 2877-2904. doi: 10.1016/j.jde.2013.01.013.
[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), 1996-2015. doi: 10.1016/j.spa.2010.06.001.
[9]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Sencond Order, Springer, 2001.
[10]	Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic differential equations, Probability Theory and Related Fields, 123 (2002), 381-411. doi: 10.1007/s004400100193.
[11]	Y. Hu and S. Peng, Adapted solution of a backward semi-linear stochastic evolution equations, Stochastic Analysis and Applications, 9 (1991), 445-459. doi: 10.1080/07362999108809250.
[12]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968.
[13]	J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stochastic Processes and their Applications, 70 (1997), 59-84. doi: 10.1016/S0304-4149(97)00057-4.
[14]	J. Ma and J. Yong, On linear, degenerate backward stochastic differential equations, Probability Theory and Related Fields, 113 (1999), 135-170. doi: 10.1007/s004400050205.
[15]	R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Hölder classes, Annals of Probability, 28 (2000), 74-103. doi: 10.1214/aop/1019160112.
[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), 185-222. doi: 10.1016/S0304-4149(03)00042-5.
[17]	S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 30 (1992), 284-304. doi: 10.1137/0330018.
[18]	J. Qiu and S. Tang, Maximum principles for backward stochastic partial differential equations, Journal of Functional Analysis, 262 (2012), 2436-2480. doi: 10.1016/j.jfa.2011.12.002.
[19]	J. Qiu, S. Tang and Y. You, 2D backward stochastic Navier-Stokes equations with nonlinear forcing, Stochastic Processes and their Applications, 122 (2012), 334-356. doi: 10.1016/j.spa.2011.08.010.
[20]	B. Rozovskiĭ, On stochastic partial differential equations, Sbornik: Mathematics, 25 (1975), 295-322.
[21]	S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations, SIAM J. Control Optim., 36 (1998), 1596-1617. doi: 10.1137/S0363012996313100.
[22]	S. Tang, A new partially observed stochastic maximum principle, in 37th IEEE Control and Decision Conference, Tampa, Florida, 1998, 2353-2358.
[23]	S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$, Chinese Annals of Mathematics, 26 (2005), 437-456. doi: 10.1142/S025295990500035X.
[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, ().
[25]	X. Zhou, A duality analysis on stochastic partial differential equations, Journal of Functional Analysis, 103 (1992), 275-293. doi: 10.1016/0022-1236(92)90122-Y.
[26]	X. Zhou, On the necessary condition of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1993), 1462-1478. doi: 10.1137/0331068.

show all references

References:

[1]	A. Bensoussan, Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions, Stochastics, 9 (1983), 169-222. doi: 10.1080/17442508308833253.
[2]	A. Bensoussan, Stochastic Control of Partially Observed Systems, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511526503.
[3]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.
[4]	N. Dokuchaev, Backward parabolic Itô equations and the second fundamental inequality, Random Operators and Stochastic Equations, 20 (2012), 69-102. doi: 10.1515/rose-2012-0003.
[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), 175-219. doi: 10.1007/s00245-011-9154-9.
[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), 255-285. doi: 10.1007/s00440-011-0369-0.
[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), 2877-2904. doi: 10.1016/j.jde.2013.01.013.
[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), 1996-2015. doi: 10.1016/j.spa.2010.06.001.
[9]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Sencond Order, Springer, 2001.
[10]	Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic differential equations, Probability Theory and Related Fields, 123 (2002), 381-411. doi: 10.1007/s004400100193.
[11]	Y. Hu and S. Peng, Adapted solution of a backward semi-linear stochastic evolution equations, Stochastic Analysis and Applications, 9 (1991), 445-459. doi: 10.1080/07362999108809250.
[12]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, 1968.
[13]	J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stochastic Processes and their Applications, 70 (1997), 59-84. doi: 10.1016/S0304-4149(97)00057-4.
[14]	J. Ma and J. Yong, On linear, degenerate backward stochastic differential equations, Probability Theory and Related Fields, 113 (1999), 135-170. doi: 10.1007/s004400050205.
[15]	R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Hölder classes, Annals of Probability, 28 (2000), 74-103. doi: 10.1214/aop/1019160112.
[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), 185-222. doi: 10.1016/S0304-4149(03)00042-5.
[17]	S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 30 (1992), 284-304. doi: 10.1137/0330018.
[18]	J. Qiu and S. Tang, Maximum principles for backward stochastic partial differential equations, Journal of Functional Analysis, 262 (2012), 2436-2480. doi: 10.1016/j.jfa.2011.12.002.
[19]	J. Qiu, S. Tang and Y. You, 2D backward stochastic Navier-Stokes equations with nonlinear forcing, Stochastic Processes and their Applications, 122 (2012), 334-356. doi: 10.1016/j.spa.2011.08.010.
[20]	B. Rozovskiĭ, On stochastic partial differential equations, Sbornik: Mathematics, 25 (1975), 295-322.
[21]	S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations, SIAM J. Control Optim., 36 (1998), 1596-1617. doi: 10.1137/S0363012996313100.
[22]	S. Tang, A new partially observed stochastic maximum principle, in 37th IEEE Control and Decision Conference, Tampa, Florida, 1998, 2353-2358.
[23]	S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$, Chinese Annals of Mathematics, 26 (2005), 437-456. doi: 10.1142/S025295990500035X.
[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, ().
[25]	X. Zhou, A duality analysis on stochastic partial differential equations, Journal of Functional Analysis, 103 (1992), 275-293. doi: 10.1016/0022-1236(92)90122-Y.
[26]	X. Zhou, On the necessary condition of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1993), 1462-1478. doi: 10.1137/0331068.

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