October  2013, 18(8): 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

Mean-square convergence of numerical approximations for a class of backward stochastic differential equations

1. 

State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China, China

Received  December 2012 Revised  April 2013 Published  July 2013

This paper is devoted to the fundamental convergence theorem on the mean-square order of numerical approximations for a class of backward stochastic differential equations with terminal condition $\chi=\varphi(W_{T}+x)$. Our theorem shows that the mean-square order of convergence of a numerical method depends on the order of the one-step approximation for the mean-square deviation only. And some numerical schemes as examples are presented to verify the theorem.
Citation: 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
References:
[1]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Math. Finance, 7 (1997), 1.  doi: 10.1111/1467-9965.00022.  Google Scholar

[2]

Y. Li and W. Zhao, $L^p$-error estimates for numerical schemes for solving certain kinds of backward stochastic differential equations,, Statist. Probab. Lett., 80 (2010), 1612.  doi: 10.1016/j.spl.2010.06.015.  Google Scholar

[3]

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

[4]

J. Ma and J. Yong, "Forward-Backward Stochastic Differential Equaitons and their Applications,", Lecture Notes in Mathematics, 1702 (1999).   Google Scholar

[5]

G. N. Milstein, "Numerical Integration of Stochastic Differential Equations,", Mathematics and its Applications, 313 (1995).   Google Scholar

[6]

É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[7]

É. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations,, in, 176 (1992), 200.  doi: 10.1007/BFb0007334.  Google Scholar

[8]

J. Wang, C. Luo and W. Zhao, Crank-Nicolson scheme and its error estimates for backward stochastic differential equations,, Acta Math. Appl. Sinica English Ser., (2009).  doi: 10.1007/s10255-009-9051-z.  Google Scholar

[9]

W. Zhao, L. Chen and S. Peng, A new kind of accurate numerical method for backward stochastic differential equations,, SIAM J. Sci. Comput., 28 (2006), 1563.  doi: 10.1137/05063341X.  Google Scholar

[10]

W. Zhao, Y. Li and G. Zhang, A generalized $\theta$-scheme for solving backward stochastic differential equations,, Discrete Contin. Dyn. Syst. B, 17 (2012), 1585.   Google Scholar

[11]

W. Zhao, J. Wang and S. Peng, Error estimates of the $\theta$-scheme for backward stochastic differential equations,, Discrete Contin. Dyn. Syst. B, 12 (2009), 905.  doi: 10.3934/dcdsb.2009.12.905.  Google Scholar

show all references

References:
[1]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Math. Finance, 7 (1997), 1.  doi: 10.1111/1467-9965.00022.  Google Scholar

[2]

Y. Li and W. Zhao, $L^p$-error estimates for numerical schemes for solving certain kinds of backward stochastic differential equations,, Statist. Probab. Lett., 80 (2010), 1612.  doi: 10.1016/j.spl.2010.06.015.  Google Scholar

[3]

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

[4]

J. Ma and J. Yong, "Forward-Backward Stochastic Differential Equaitons and their Applications,", Lecture Notes in Mathematics, 1702 (1999).   Google Scholar

[5]

G. N. Milstein, "Numerical Integration of Stochastic Differential Equations,", Mathematics and its Applications, 313 (1995).   Google Scholar

[6]

É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[7]

É. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations,, in, 176 (1992), 200.  doi: 10.1007/BFb0007334.  Google Scholar

[8]

J. Wang, C. Luo and W. Zhao, Crank-Nicolson scheme and its error estimates for backward stochastic differential equations,, Acta Math. Appl. Sinica English Ser., (2009).  doi: 10.1007/s10255-009-9051-z.  Google Scholar

[9]

W. Zhao, L. Chen and S. Peng, A new kind of accurate numerical method for backward stochastic differential equations,, SIAM J. Sci. Comput., 28 (2006), 1563.  doi: 10.1137/05063341X.  Google Scholar

[10]

W. Zhao, Y. Li and G. Zhang, A generalized $\theta$-scheme for solving backward stochastic differential equations,, Discrete Contin. Dyn. Syst. B, 17 (2012), 1585.   Google Scholar

[11]

W. Zhao, J. Wang and S. Peng, Error estimates of the $\theta$-scheme for backward stochastic differential equations,, Discrete Contin. Dyn. Syst. B, 12 (2009), 905.  doi: 10.3934/dcdsb.2009.12.905.  Google Scholar

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