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 and 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), 59-79. doi: 10.1093/rfs/11.1.59.

[2]

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.

[3]

B. Bouchard and R. Elie, Discrete-time approximation of decoupled forward-backward SDE with jumps, Stochastic Process. Appl., 118 (2008), 53-75. doi: 10.1016/j.spa.2007.03.010.

[4]

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111 (2004), 175-206.

[5]

D. Chevance, Numerical methods for backward stochastic differential equations, "Numerical Methods in Finance,'' Publ. Newton Inst., Cambridge Univ. Press, Cambridge, (1997), 232-244.

[6]

J. Cvitanić and J. Zhang, The steepest descent method for forward-backward SDEs, Electron. J. Probab., 10 (2005), 1468-1495.

[7]

F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab., 16 (2006), 140-184. doi: 10.1214/105051605000000674.

[8]

J. Douglas, Jr., J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968.

[9]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, R.I., 1998.

[10]

E. R. Gianin, Risk measures via g-expectations, Insurance Math. Econom., 39 (2006), 19-34. doi: 10.1016/j.insmatheco.2006.01.002.

[11]

E. Gobet and C. Labart, Error expansion for the discretization of backward stochastic differential equations, Stochastic Process. Appl., 117 (2007), 803-829. doi: 10.1016/j.spa.2006.10.007.

[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), 2172-2202. doi: 10.1214/105051605000000412.

[13]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.

[14]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Translations of Math. Monographs, American Mathematical Society, 23, Providence, R.I., 1968.

[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), 1612-1617. doi: 10.1016/j.spl.2010.06.015.

[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), 339-359. doi: 10.1007/BF01192258.

[17]

J. Ma, J. Shen and Y. Zhao, On Numerical approximations of forward-backward stochastic differential equations, SIAM J.Numer. Anal., 46 (2008), 2636-2661. doi: 10.1137/06067393X.

[18]

J. Ma and J. Zhang, Representation theorems for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 1390-1418.

[19]

G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 561-582. doi: 10.1137/040614426.

[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), 24-44.

[21]

É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.

[22]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054.

[23]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74.

[24]

S. Peng, Backward SDE and Related g-expectation, in "Backward Stochastic Differential Equations'' (Paris, 1995-1996), Pitman Res. Notes Math. Ser., 364, Longman, Harlow, (1997), 141-159.

[25]

S. Peng, A linear approximation algorithm using BSDE, Pacific Economic Review, 4 (1999), 285-291. doi: 10.1111/1468-0106.00079.

[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.

[27]

J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab., 14 (2004), 459-488. doi: 10.1214/aoap/1075828058.

[28]

Y. Zhang and W. Zheng, Discretizing a backward stochastic differential equation, Int. J. Math. Math. Sci., 32 (2002), 103-116.

[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), 1563-1581. doi: 10.1137/05063341X.

[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), 905-924. doi: 10.3934/dcdsb.2009.12.905.

[31]

W. Zhao, G. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 1369-1394. doi: 10.1137/09076979X.

show all references

References:
[1]

M. Broadie, J. Cvitanic and H. M. Soner, Optimal replication of contingent claims under portfolio constraints, Rev. of Financial Studies, 11 (1998), 59-79. doi: 10.1093/rfs/11.1.59.

[2]

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.

[3]

B. Bouchard and R. Elie, Discrete-time approximation of decoupled forward-backward SDE with jumps, Stochastic Process. Appl., 118 (2008), 53-75. doi: 10.1016/j.spa.2007.03.010.

[4]

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111 (2004), 175-206.

[5]

D. Chevance, Numerical methods for backward stochastic differential equations, "Numerical Methods in Finance,'' Publ. Newton Inst., Cambridge Univ. Press, Cambridge, (1997), 232-244.

[6]

J. Cvitanić and J. Zhang, The steepest descent method for forward-backward SDEs, Electron. J. Probab., 10 (2005), 1468-1495.

[7]

F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab., 16 (2006), 140-184. doi: 10.1214/105051605000000674.

[8]

J. Douglas, Jr., J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968.

[9]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, R.I., 1998.

[10]

E. R. Gianin, Risk measures via g-expectations, Insurance Math. Econom., 39 (2006), 19-34. doi: 10.1016/j.insmatheco.2006.01.002.

[11]

E. Gobet and C. Labart, Error expansion for the discretization of backward stochastic differential equations, Stochastic Process. Appl., 117 (2007), 803-829. doi: 10.1016/j.spa.2006.10.007.

[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), 2172-2202. doi: 10.1214/105051605000000412.

[13]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), 1-71.

[14]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Translations of Math. Monographs, American Mathematical Society, 23, Providence, R.I., 1968.

[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), 1612-1617. doi: 10.1016/j.spl.2010.06.015.

[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), 339-359. doi: 10.1007/BF01192258.

[17]

J. Ma, J. Shen and Y. Zhao, On Numerical approximations of forward-backward stochastic differential equations, SIAM J.Numer. Anal., 46 (2008), 2636-2661. doi: 10.1137/06067393X.

[18]

J. Ma and J. Zhang, Representation theorems for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 1390-1418.

[19]

G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 561-582. doi: 10.1137/040614426.

[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), 24-44.

[21]

É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.

[22]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979. doi: 10.1137/0328054.

[23]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep., 37 (1991), 61-74.

[24]

S. Peng, Backward SDE and Related g-expectation, in "Backward Stochastic Differential Equations'' (Paris, 1995-1996), Pitman Res. Notes Math. Ser., 364, Longman, Harlow, (1997), 141-159.

[25]

S. Peng, A linear approximation algorithm using BSDE, Pacific Economic Review, 4 (1999), 285-291. doi: 10.1111/1468-0106.00079.

[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.

[27]

J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab., 14 (2004), 459-488. doi: 10.1214/aoap/1075828058.

[28]

Y. Zhang and W. Zheng, Discretizing a backward stochastic differential equation, Int. J. Math. Math. Sci., 32 (2002), 103-116.

[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), 1563-1581. doi: 10.1137/05063341X.

[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), 905-924. doi: 10.3934/dcdsb.2009.12.905.

[31]

W. Zhao, G. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 1369-1394. doi: 10.1137/09076979X.

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