-
Previous Article
Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations
- DCDS Home
- This Issue
-
Next Article
On forward and backward SPDEs with non-local boundary conditions
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 |
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. |
[1] |
Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control and Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047 |
[2] |
Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075 |
[3] |
Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803 |
[4] |
Joscha Diehl, Jianfeng Zhang. Backward stochastic differential equations with Young drift. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 5-. doi: 10.1186/s41546-017-0016-5 |
[5] |
Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002 |
[6] |
Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control and Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020 |
[7] |
Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447 |
[8] |
Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115 |
[9] |
Renzhi Qiu, Shanjian Tang. The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 3-. doi: 10.1186/s41546-019-0037-3 |
[10] |
Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 |
[11] |
Dinh Nguyen Duy Hai. Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1715-1734. doi: 10.3934/cpaa.2022043 |
[12] |
Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285 |
[13] |
Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565 |
[14] |
Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control and Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013 |
[15] |
Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905 |
[16] |
Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585 |
[17] |
Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 773-795. doi: 10.3934/dcdss.2021044 |
[18] |
Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure and Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793 |
[19] |
Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051 |
[20] |
Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]