# American Institute of Mathematical Sciences

July  2012, 17(5): 1585-1603. doi: 10.3934/dcdsb.2012.17.1585

## A generalized $\theta$-scheme for solving backward stochastic differential equations

 1 School of Mathematics, Shandong University, Jinan, Shandong, China 2 Department of Scientiﬁc Computing, Florida State University, Tallahassee, FL 32306, United States

Received  August 2011 Revised  November 2011 Published  March 2012

In this paper we propose a new type of $\theta$-scheme with four parameters ($\{\theta_i\}_{i=1}^4$) for solving the backward stochastic differential equation $-dy_t=f(t,y_t,z_t) dt - z_t dW_t$. We rigorously prove some error estimates for the proposed scheme, and in particular, we show that accuracy of the scheme can be high by choosing proper parameters. Various numerical examples are also presented to verify the theoretical results.
Citation: 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
##### References:
 [1] M. Broadie, J. Cvitanic and H. M. Soner, Optimal replication of contingent claims under portfolio constraints,, Rev. of Financial Studies, 11 (1998), 59. doi: 10.1093/rfs/11.1.59. Google Scholar [2] 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 [3] B. Bouchard and R. Elie, Discrete-time approximation of decoupled forward-backward SDE with jumps,, Stochastic Process. Appl., 118 (2008), 53. doi: 10.1016/j.spa.2007.03.010. Google Scholar [4] B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations,, Stochastic Process. Appl., 111 (2004), 175. Google Scholar [5] D. Chevance, Numerical methods for backward stochastic differential equations,, Publ. Newton Inst., (1997), 232. Google Scholar [6] J. Cvitanić and J. Zhang, The steepest descent method for forward-backward SDEs,, Electron. J. Probab., 10 (2005), 1468. Google Scholar [7] F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs,, Ann. Appl. Probab., 16 (2006), 140. doi: 10.1214/105051605000000674. Google Scholar [8] J. Douglas, Jr., J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations,, Ann. Appl. Probab., 6 (1996), 940. Google Scholar [9] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar [10] E. R. Gianin, Risk measures via g-expectations,, Insurance Math. Econom., 39 (2006), 19. doi: 10.1016/j.insmatheco.2006.01.002. Google Scholar [11] E. Gobet and C. Labart, Error expansion for the discretization of backward stochastic differential equations,, Stochastic Process. Appl., 117 (2007), 803. doi: 10.1016/j.spa.2006.10.007. Google Scholar [12] E. Gobet, J.-P. Lemmor 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 [13] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Mathematical Finance, 7 (1997), 1. Google Scholar [14] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,", Translations of Math. Monographs, 23 (1968). Google Scholar [15] Y. Li and W. Zhao, $L^p$-error estimates for numerical schemes for solving certain kinds of backward stochastic differential equations,, Statistics and Probability Letters, 80 (2010), 1612. doi: 10.1016/j.spl.2010.06.015. Google Scholar [16] J. Ma, P. Protter and J. 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 [17] J. Ma, J. Shen and Y. Zhao, On Numerical approximations of forward-backward stochastic differential equations,, SIAM J.Numer. Anal., 46 (2008), 2636. doi: 10.1137/06067393X. Google Scholar [18] J. Ma and J. Zhang, Representation theorems for backward stochastic differential equations,, Ann. Appl. Probab., 12 (2002), 1390. 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] G. N. Milstein and M. V. Tretyakov, Discretization of forward-backward stochastic differential equations and related quasi-linear parabolic equations,, IMA J. Numer. Anal., 27 (2007), 24. Google Scholar [21] 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 [22] S. Peng, A general stochastic maximum principle for optimal control problems,, SIAM J. Control Optim., 28 (1990), 966. doi: 10.1137/0328054. Google Scholar [23] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,, Stochastics Stochastics Rep., 37 (1991), 61. Google Scholar [24] S. Peng, Backward SDE and Related g-expectation,, in, 364 (1997), 1995. Google Scholar [25] S. Peng, A linear approximation algorithm using BSDE,, Pacific Economic Review, 4 (1999), 285. doi: 10.1111/1468-0106.00079. Google Scholar [26] J. Wang, C. Luo and W. Zhao, Crank-Nicolson scheme and its error estimates for backward stochastic differential equations,, Acta Mathematicae Applicatae Sinica (English Series), (2009). Google Scholar [27] J. Zhang, A numerical scheme for BSDEs,, Ann. Appl. Probab., 14 (2004), 459. doi: 10.1214/aoap/1075828058. Google Scholar [28] Y. Zhang and W. Zheng, Discretizing a backward stochastic differential equation,, Int. J. Math. Math. Sci., 32 (2002), 103. Google Scholar [29] 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 [30] W. Zhao, J. Wang and S. Peng, Error estimates of the $\theta$-scheme for backward stochastic differential equations,, Dis. Cont. Dyn. Sys. B, 12 (2009), 905. doi: 10.3934/dcdsb.2009.12.905. Google Scholar [31] W. Zhao, G. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations,, SIAM J. Numer. Anal., 48 (2010), 1369. doi: 10.1137/09076979X. Google Scholar

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##### References:
 [1] M. Broadie, J. Cvitanic and H. M. Soner, Optimal replication of contingent claims under portfolio constraints,, Rev. of Financial Studies, 11 (1998), 59. doi: 10.1093/rfs/11.1.59. Google Scholar [2] 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 [3] B. Bouchard and R. Elie, Discrete-time approximation of decoupled forward-backward SDE with jumps,, Stochastic Process. Appl., 118 (2008), 53. doi: 10.1016/j.spa.2007.03.010. Google Scholar [4] B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations,, Stochastic Process. Appl., 111 (2004), 175. Google Scholar [5] D. Chevance, Numerical methods for backward stochastic differential equations,, Publ. Newton Inst., (1997), 232. Google Scholar [6] J. Cvitanić and J. Zhang, The steepest descent method for forward-backward SDEs,, Electron. J. Probab., 10 (2005), 1468. Google Scholar [7] F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs,, Ann. Appl. Probab., 16 (2006), 140. doi: 10.1214/105051605000000674. Google Scholar [8] J. Douglas, Jr., J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations,, Ann. Appl. Probab., 6 (1996), 940. Google Scholar [9] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar [10] E. R. Gianin, Risk measures via g-expectations,, Insurance Math. Econom., 39 (2006), 19. doi: 10.1016/j.insmatheco.2006.01.002. Google Scholar [11] E. Gobet and C. Labart, Error expansion for the discretization of backward stochastic differential equations,, Stochastic Process. Appl., 117 (2007), 803. doi: 10.1016/j.spa.2006.10.007. Google Scholar [12] E. Gobet, J.-P. Lemmor 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 [13] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Mathematical Finance, 7 (1997), 1. Google Scholar [14] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,", Translations of Math. Monographs, 23 (1968). Google Scholar [15] Y. Li and W. Zhao, $L^p$-error estimates for numerical schemes for solving certain kinds of backward stochastic differential equations,, Statistics and Probability Letters, 80 (2010), 1612. doi: 10.1016/j.spl.2010.06.015. Google Scholar [16] J. Ma, P. Protter and J. 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 [17] J. Ma, J. Shen and Y. Zhao, On Numerical approximations of forward-backward stochastic differential equations,, SIAM J.Numer. Anal., 46 (2008), 2636. doi: 10.1137/06067393X. Google Scholar [18] J. Ma and J. Zhang, Representation theorems for backward stochastic differential equations,, Ann. Appl. Probab., 12 (2002), 1390. 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] G. N. Milstein and M. V. Tretyakov, Discretization of forward-backward stochastic differential equations and related quasi-linear parabolic equations,, IMA J. Numer. Anal., 27 (2007), 24. Google Scholar [21] 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 [22] S. Peng, A general stochastic maximum principle for optimal control problems,, SIAM J. Control Optim., 28 (1990), 966. doi: 10.1137/0328054. Google Scholar [23] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,, Stochastics Stochastics Rep., 37 (1991), 61. Google Scholar [24] S. Peng, Backward SDE and Related g-expectation,, in, 364 (1997), 1995. Google Scholar [25] S. Peng, A linear approximation algorithm using BSDE,, Pacific Economic Review, 4 (1999), 285. doi: 10.1111/1468-0106.00079. Google Scholar [26] J. Wang, C. Luo and W. Zhao, Crank-Nicolson scheme and its error estimates for backward stochastic differential equations,, Acta Mathematicae Applicatae Sinica (English Series), (2009). Google Scholar [27] J. Zhang, A numerical scheme for BSDEs,, Ann. Appl. Probab., 14 (2004), 459. doi: 10.1214/aoap/1075828058. Google Scholar [28] Y. Zhang and W. Zheng, Discretizing a backward stochastic differential equation,, Int. J. Math. Math. Sci., 32 (2002), 103. Google Scholar [29] 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 [30] W. Zhao, J. Wang and S. Peng, Error estimates of the $\theta$-scheme for backward stochastic differential equations,, Dis. Cont. Dyn. Sys. B, 12 (2009), 905. doi: 10.3934/dcdsb.2009.12.905. Google Scholar [31] W. Zhao, G. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations,, SIAM J. Numer. Anal., 48 (2010), 1369. doi: 10.1137/09076979X. Google Scholar
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