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A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations
1. | School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
References:
[1] |
C. Bender and R. Denk, A forward scheme for backward SDEs, Stochastic Process. Appl., 117 (2007), 1793-1812.
doi: 10.1016/j.spa.2007.03.005. |
[2] |
B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111 (2004), 175-206.
doi: 10.1016/j.spa.2004.01.001. |
[3] |
P. Briand and C. Labart, Simulation of BSDEs by Wiener chaos expansion, Ann. Appl. Probab., 24 (2014), 1129-1171.
doi: 10.1214/13-AAP943. |
[4] |
J. Douglas Jr., J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968.
doi: 10.1214/aoap/1034968235. |
[5] |
M. Fuhrman and Y. Hu, Infinite horizon BSDEs in infinite dimensions with continuous driver and applications, J. Evol. Equ., 6 (2006), 459-484.
doi: 10.1007/s00028-006-0263-x. |
[6] |
E. Gobet, J.-P. Lemor and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab., 15 (2005), 2172-2202.
doi: 10.1214/105051605000000412. |
[7] |
W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Austral. Math. Soc., 54 (1996), 79-85.
doi: 10.1017/S0004972700015094. |
[8] |
I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I, Potential Anal., 9 (1998), 1-25.
doi: 10.1023/A:1008615012377. |
[9] |
T. Y. Hou, W. Luo, B. Rozovskii and H.-M. Zhou, Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics, J. Comput. Phys., 216 (2006), 687-706.
doi: 10.1016/j.jcp.2006.01.008. |
[10] |
Y. Hu, D. Nualart and X. Song, Malliavin calculus for backward stochastic differential equations and application to numerical solutions, Ann. Appl. Probab., 21 (2011), 2379-2423.
doi: 10.1214/11-AAP762. |
[11] |
Y. Hu and S. G. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 33 (1990), 159-180.
doi: 10.1080/17442509008833671. |
[12] |
Y. Hu and S. G. Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Anal. Appl., 9 (1991), 445-459.
doi: 10.1080/07362999108809250. |
[13] |
A. Jentzen and P. E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 649-667.
doi: 10.1098/rspa.2008.0325. |
[14] |
S. Lototsky, R. Mikulevicius and B. L. Rozovskii, Nonlinear filtering revisited: A spectral approach, SIAM J. Control Optim., 35 (1997), 435-461.
doi: 10.1137/S0363012993248918. |
[15] |
Q. Lü and X. Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, SpringerBriefs in Mathematics, Springer, Cham, 2014.
doi: 10.1007/978-3-319-06632-5. |
[16] |
J. Ma, P. Protter, J. San Martin and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 302-316.
doi: 10.1214/aoap/1015961165. |
[17] |
J. Ma, P. Protter and J. M. Yong, Solving forward-backward stochastic differential equations explicitly-a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359.
doi: 10.1007/BF01192258. |
[18] |
J. Ma and J. Zhang, Path regularity for solutions of backward stochastic differential equations, Probab. Theory Related Fields, 122 (2002), 163-190.
doi: 10.1007/s004400100144. |
[19] |
G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 561-582 (electronic).
doi: 10.1137/040614426. |
[20] |
E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167.
doi: 10.1080/17442507908833142. |
[21] |
T. Shardlow, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim., 20 (1999), 121-145.
doi: 10.1080/01630569908816884. |
[22] |
G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE, Stochastic Anal. Appl., 14 (1996), 461-486.
doi: 10.1080/07362999608809451. |
[23] |
J. B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Anal., 23 (2005), 1-43.
doi: 10.1007/s11118-004-2950-y. |
[24] |
P. Wang and X. Zhang, Numerical solutions of backward stochastic differential equations: A finite transposition method, C. R. Math. Acad. Sci. Paris, 349 (2011), 901-903.
doi: 10.1016/j.crma.2011.07.011. |
[25] |
Y. Wang, Transposition Solutions of Backward Stochastic Differential Equations and Numerical Schemes,, 2013, ().
|
[26] |
Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise, BIT, 44 (2004), 829-847.
doi: 10.1007/s10543-004-3755-5. |
[27] |
A. N. Yannacopoulos, N. E. Frangos and I. Karatzas, Wiener chaos solutions for linear backward stochastic evolution equations, SIAM J. Math. Anal., 43 (2011), 68-113.
doi: 10.1137/090750652. |
[28] |
J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, vol. 43 of Applications of Mathematics (New York), Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
[29] |
J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab., 14 (2004), 459-488.
doi: 10.1214/aoap/1075828058. |
[30] |
X. Y. Zhou, A duality analysis on stochastic partial differential equations, J. Funct. Anal., 103 (1992), 275-293.
doi: 10.1016/0022-1236(92)90122-Y. |
show all references
References:
[1] |
C. Bender and R. Denk, A forward scheme for backward SDEs, Stochastic Process. Appl., 117 (2007), 1793-1812.
doi: 10.1016/j.spa.2007.03.005. |
[2] |
B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111 (2004), 175-206.
doi: 10.1016/j.spa.2004.01.001. |
[3] |
P. Briand and C. Labart, Simulation of BSDEs by Wiener chaos expansion, Ann. Appl. Probab., 24 (2014), 1129-1171.
doi: 10.1214/13-AAP943. |
[4] |
J. Douglas Jr., J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968.
doi: 10.1214/aoap/1034968235. |
[5] |
M. Fuhrman and Y. Hu, Infinite horizon BSDEs in infinite dimensions with continuous driver and applications, J. Evol. Equ., 6 (2006), 459-484.
doi: 10.1007/s00028-006-0263-x. |
[6] |
E. Gobet, J.-P. Lemor and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab., 15 (2005), 2172-2202.
doi: 10.1214/105051605000000412. |
[7] |
W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs, Bull. Austral. Math. Soc., 54 (1996), 79-85.
doi: 10.1017/S0004972700015094. |
[8] |
I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I, Potential Anal., 9 (1998), 1-25.
doi: 10.1023/A:1008615012377. |
[9] |
T. Y. Hou, W. Luo, B. Rozovskii and H.-M. Zhou, Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics, J. Comput. Phys., 216 (2006), 687-706.
doi: 10.1016/j.jcp.2006.01.008. |
[10] |
Y. Hu, D. Nualart and X. Song, Malliavin calculus for backward stochastic differential equations and application to numerical solutions, Ann. Appl. Probab., 21 (2011), 2379-2423.
doi: 10.1214/11-AAP762. |
[11] |
Y. Hu and S. G. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep., 33 (1990), 159-180.
doi: 10.1080/17442509008833671. |
[12] |
Y. Hu and S. G. Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Anal. Appl., 9 (1991), 445-459.
doi: 10.1080/07362999108809250. |
[13] |
A. Jentzen and P. E. Kloeden, Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 649-667.
doi: 10.1098/rspa.2008.0325. |
[14] |
S. Lototsky, R. Mikulevicius and B. L. Rozovskii, Nonlinear filtering revisited: A spectral approach, SIAM J. Control Optim., 35 (1997), 435-461.
doi: 10.1137/S0363012993248918. |
[15] |
Q. Lü and X. Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions, SpringerBriefs in Mathematics, Springer, Cham, 2014.
doi: 10.1007/978-3-319-06632-5. |
[16] |
J. Ma, P. Protter, J. San Martin and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 302-316.
doi: 10.1214/aoap/1015961165. |
[17] |
J. Ma, P. Protter and J. M. Yong, Solving forward-backward stochastic differential equations explicitly-a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359.
doi: 10.1007/BF01192258. |
[18] |
J. Ma and J. Zhang, Path regularity for solutions of backward stochastic differential equations, Probab. Theory Related Fields, 122 (2002), 163-190.
doi: 10.1007/s004400100144. |
[19] |
G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 561-582 (electronic).
doi: 10.1137/040614426. |
[20] |
E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167.
doi: 10.1080/17442507908833142. |
[21] |
T. Shardlow, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim., 20 (1999), 121-145.
doi: 10.1080/01630569908816884. |
[22] |
G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE, Stochastic Anal. Appl., 14 (1996), 461-486.
doi: 10.1080/07362999608809451. |
[23] |
J. B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Anal., 23 (2005), 1-43.
doi: 10.1007/s11118-004-2950-y. |
[24] |
P. Wang and X. Zhang, Numerical solutions of backward stochastic differential equations: A finite transposition method, C. R. Math. Acad. Sci. Paris, 349 (2011), 901-903.
doi: 10.1016/j.crma.2011.07.011. |
[25] |
Y. Wang, Transposition Solutions of Backward Stochastic Differential Equations and Numerical Schemes,, 2013, ().
|
[26] |
Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise, BIT, 44 (2004), 829-847.
doi: 10.1007/s10543-004-3755-5. |
[27] |
A. N. Yannacopoulos, N. E. Frangos and I. Karatzas, Wiener chaos solutions for linear backward stochastic evolution equations, SIAM J. Math. Anal., 43 (2011), 68-113.
doi: 10.1137/090750652. |
[28] |
J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, vol. 43 of Applications of Mathematics (New York), Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
[29] |
J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab., 14 (2004), 459-488.
doi: 10.1214/aoap/1075828058. |
[30] |
X. Y. Zhou, A duality analysis on stochastic partial differential equations, J. Funct. Anal., 103 (1992), 275-293.
doi: 10.1016/0022-1236(92)90122-Y. |
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