September  2016, 6(3): 489-515. doi: 10.3934/mcrf.2016013

A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  May 2015 Revised  January 2016 Published  August 2016

In this paper, we present a numerical scheme to solve the initial-boundary value problem for backward stochastic partial differential equations of parabolic type. Based on the Galerkin method, we approximate the original equation by a family of backward stochastic differential equations (BSDEs, for short), and then solve these BSDEs by the time discretization. Combining the truncation with respect to the spatial variable and the backward Euler method on time variable, we obtain the global $L^2$ error estimate.
Citation: 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
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, Thesis (Ph.D.)-Academy of Mathematics and Systems Science, Chinese Academy of Sciences.

[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, Thesis (Ph.D.)-Academy of Mathematics and Systems Science, Chinese Academy of Sciences.

[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.

[1]

Martin Hutzenthaler, Thomas Kruse, Tuan Anh Nguyen. On the speed of convergence of Picard iterations of backward stochastic differential equations. Probability, Uncertainty and Quantitative Risk, 2022, 7 (2) : 133-150. doi: 10.3934/puqr.2022009

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065

[10]

Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations and Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017

[11]

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

[12]

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

[13]

Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028

[14]

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

[15]

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

[16]

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

[17]

Jiongmin Yong. Forward-backward stochastic differential equations: Initiation, development and beyond. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022011

[18]

Yinggu Chen, Said HamadÈne, Tingshu Mu. Mean-field doubly reflected backward stochastic differential equations. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022012

[19]

Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic and Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044

[20]

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

2021 Impact Factor: 1.141

Metrics

  • PDF downloads (221)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]