# American Institute of Mathematical Sciences

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, (). [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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##### 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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