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 & 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.  doi: 10.1016/j.spa.2007.03.005.  Google Scholar

[2]

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations,, Stochastic Process. Appl., 111 (2004), 175.  doi: 10.1016/j.spa.2004.01.001.  Google Scholar

[3]

P. Briand and C. Labart, Simulation of BSDEs by Wiener chaos expansion,, Ann. Appl. Probab., 24 (2014), 1129.  doi: 10.1214/13-AAP943.  Google Scholar

[4]

J. Douglas Jr., J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations,, Ann. Appl. Probab., 6 (1996), 940.  doi: 10.1214/aoap/1034968235.  Google Scholar

[5]

M. Fuhrman and Y. Hu, Infinite horizon BSDEs in infinite dimensions with continuous driver and applications,, J. Evol. Equ., 6 (2006), 459.  doi: 10.1007/s00028-006-0263-x.  Google Scholar

[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.  doi: 10.1214/105051605000000412.  Google Scholar

[7]

W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs,, Bull. Austral. Math. Soc., 54 (1996), 79.  doi: 10.1017/S0004972700015094.  Google Scholar

[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.  doi: 10.1023/A:1008615012377.  Google Scholar

[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.  doi: 10.1016/j.jcp.2006.01.008.  Google Scholar

[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.  doi: 10.1214/11-AAP762.  Google Scholar

[11]

Y. Hu and S. G. Peng, Maximum principle for semilinear stochastic evolution control systems,, Stochastics Stochastics Rep., 33 (1990), 159.  doi: 10.1080/17442509008833671.  Google Scholar

[12]

Y. Hu and S. G. Peng, Adapted solution of a backward semilinear stochastic evolution equation,, Stochastic Anal. Appl., 9 (1991), 445.  doi: 10.1080/07362999108809250.  Google Scholar

[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.  doi: 10.1098/rspa.2008.0325.  Google Scholar

[14]

S. Lototsky, R. Mikulevicius and B. L. Rozovskii, Nonlinear filtering revisited: A spectral approach,, SIAM J. Control Optim., 35 (1997), 435.  doi: 10.1137/S0363012993248918.  Google Scholar

[15]

Q. Lü and X. Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions,, SpringerBriefs in Mathematics, (2014).  doi: 10.1007/978-3-319-06632-5.  Google Scholar

[16]

J. Ma, P. Protter, J. San Martin and S. Torres, Numerical method for backward stochastic differential equations,, Ann. Appl. Probab., 12 (2002), 302.  doi: 10.1214/aoap/1015961165.  Google Scholar

[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.  doi: 10.1007/BF01192258.  Google Scholar

[18]

J. Ma and J. Zhang, Path regularity for solutions of backward stochastic differential equations,, Probab. Theory Related Fields, 122 (2002), 163.  doi: 10.1007/s004400100144.  Google Scholar

[19]

G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward-backward stochastic differential equations,, SIAM J. Sci. Comput., 28 (2006), 561.  doi: 10.1137/040614426.  Google Scholar

[20]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes,, Stochastics, 3 (1979), 127.  doi: 10.1080/17442507908833142.  Google Scholar

[21]

T. Shardlow, Numerical methods for stochastic parabolic PDEs,, Numer. Funct. Anal. Optim., 20 (1999), 121.  doi: 10.1080/01630569908816884.  Google Scholar

[22]

G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE,, Stochastic Anal. Appl., 14 (1996), 461.  doi: 10.1080/07362999608809451.  Google Scholar

[23]

J. B. Walsh, Finite element methods for parabolic stochastic PDE's,, Potential Anal., 23 (2005), 1.  doi: 10.1007/s11118-004-2950-y.  Google Scholar

[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.  doi: 10.1016/j.crma.2011.07.011.  Google Scholar

[25]

Y. Wang, Transposition Solutions of Backward Stochastic Differential Equations and Numerical Schemes,, 2013, ().   Google Scholar

[26]

Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise,, BIT, 44 (2004), 829.  doi: 10.1007/s10543-004-3755-5.  Google Scholar

[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.  doi: 10.1137/090750652.  Google Scholar

[28]

J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, vol. 43 of Applications of Mathematics (New York),, Springer-Verlag, (1999).  doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[29]

J. Zhang, A numerical scheme for BSDEs,, Ann. Appl. Probab., 14 (2004), 459.  doi: 10.1214/aoap/1075828058.  Google Scholar

[30]

X. Y. Zhou, A duality analysis on stochastic partial differential equations,, J. Funct. Anal., 103 (1992), 275.  doi: 10.1016/0022-1236(92)90122-Y.  Google Scholar

show all references

References:
[1]

C. Bender and R. Denk, A forward scheme for backward SDEs,, Stochastic Process. Appl., 117 (2007), 1793.  doi: 10.1016/j.spa.2007.03.005.  Google Scholar

[2]

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations,, Stochastic Process. Appl., 111 (2004), 175.  doi: 10.1016/j.spa.2004.01.001.  Google Scholar

[3]

P. Briand and C. Labart, Simulation of BSDEs by Wiener chaos expansion,, Ann. Appl. Probab., 24 (2014), 1129.  doi: 10.1214/13-AAP943.  Google Scholar

[4]

J. Douglas Jr., J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations,, Ann. Appl. Probab., 6 (1996), 940.  doi: 10.1214/aoap/1034968235.  Google Scholar

[5]

M. Fuhrman and Y. Hu, Infinite horizon BSDEs in infinite dimensions with continuous driver and applications,, J. Evol. Equ., 6 (2006), 459.  doi: 10.1007/s00028-006-0263-x.  Google Scholar

[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.  doi: 10.1214/105051605000000412.  Google Scholar

[7]

W. Grecksch and P. E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs,, Bull. Austral. Math. Soc., 54 (1996), 79.  doi: 10.1017/S0004972700015094.  Google Scholar

[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.  doi: 10.1023/A:1008615012377.  Google Scholar

[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.  doi: 10.1016/j.jcp.2006.01.008.  Google Scholar

[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.  doi: 10.1214/11-AAP762.  Google Scholar

[11]

Y. Hu and S. G. Peng, Maximum principle for semilinear stochastic evolution control systems,, Stochastics Stochastics Rep., 33 (1990), 159.  doi: 10.1080/17442509008833671.  Google Scholar

[12]

Y. Hu and S. G. Peng, Adapted solution of a backward semilinear stochastic evolution equation,, Stochastic Anal. Appl., 9 (1991), 445.  doi: 10.1080/07362999108809250.  Google Scholar

[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.  doi: 10.1098/rspa.2008.0325.  Google Scholar

[14]

S. Lototsky, R. Mikulevicius and B. L. Rozovskii, Nonlinear filtering revisited: A spectral approach,, SIAM J. Control Optim., 35 (1997), 435.  doi: 10.1137/S0363012993248918.  Google Scholar

[15]

Q. Lü and X. Zhang, General Pontryagin-type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions,, SpringerBriefs in Mathematics, (2014).  doi: 10.1007/978-3-319-06632-5.  Google Scholar

[16]

J. Ma, P. Protter, J. San Martin and S. Torres, Numerical method for backward stochastic differential equations,, Ann. Appl. Probab., 12 (2002), 302.  doi: 10.1214/aoap/1015961165.  Google Scholar

[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.  doi: 10.1007/BF01192258.  Google Scholar

[18]

J. Ma and J. Zhang, Path regularity for solutions of backward stochastic differential equations,, Probab. Theory Related Fields, 122 (2002), 163.  doi: 10.1007/s004400100144.  Google Scholar

[19]

G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward-backward stochastic differential equations,, SIAM J. Sci. Comput., 28 (2006), 561.  doi: 10.1137/040614426.  Google Scholar

[20]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes,, Stochastics, 3 (1979), 127.  doi: 10.1080/17442507908833142.  Google Scholar

[21]

T. Shardlow, Numerical methods for stochastic parabolic PDEs,, Numer. Funct. Anal. Optim., 20 (1999), 121.  doi: 10.1080/01630569908816884.  Google Scholar

[22]

G. Tessitore, Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE,, Stochastic Anal. Appl., 14 (1996), 461.  doi: 10.1080/07362999608809451.  Google Scholar

[23]

J. B. Walsh, Finite element methods for parabolic stochastic PDE's,, Potential Anal., 23 (2005), 1.  doi: 10.1007/s11118-004-2950-y.  Google Scholar

[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.  doi: 10.1016/j.crma.2011.07.011.  Google Scholar

[25]

Y. Wang, Transposition Solutions of Backward Stochastic Differential Equations and Numerical Schemes,, 2013, ().   Google Scholar

[26]

Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise,, BIT, 44 (2004), 829.  doi: 10.1007/s10543-004-3755-5.  Google Scholar

[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.  doi: 10.1137/090750652.  Google Scholar

[28]

J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, vol. 43 of Applications of Mathematics (New York),, Springer-Verlag, (1999).  doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[29]

J. Zhang, A numerical scheme for BSDEs,, Ann. Appl. Probab., 14 (2004), 459.  doi: 10.1214/aoap/1075828058.  Google Scholar

[30]

X. Y. Zhou, A duality analysis on stochastic partial differential equations,, J. Funct. Anal., 103 (1992), 275.  doi: 10.1016/0022-1236(92)90122-Y.  Google Scholar

[1]

Xiao-Qian Jiang, Lun-Chuan Zhang. A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-978. doi: 10.3934/dcdss.2019065

[2]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002

[3]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075

[4]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803

[5]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447

[6]

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

[7]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

[8]

Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285

[9]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565

[10]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905

[11]

Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585

[12]

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

[13]

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 & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017

[14]

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

[15]

Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297

[16]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[17]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501

[18]

Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control & Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013

[19]

Qi Lü, Xu Zhang. Transposition method for backward stochastic evolution equations revisited, and its application. Mathematical Control & Related Fields, 2015, 5 (3) : 529-555. doi: 10.3934/mcrf.2015.5.529

[20]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]