A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations

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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

Abstract
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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.

Keywords: backward stochastic differential equation, Backward stochastic parabolic differential equation, strong convergence., Galerkin method.

Mathematics Subject Classification: Primary: 60H15, 65M60; Secondary: 65C3.

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

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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

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