July  2015, 20(5): 1297-1313. doi: 10.3934/dcdsb.2015.20.1297

A first order semi-discrete algorithm for backward doubly stochastic differential equations

1. 

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, United States

2. 

Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849

3. 

School of Mathematics, Shandong University, Jinan, Shandong

Received  July 2013 Revised  January 2015 Published  May 2015

Numerical solutions of backward doubly stochastic differential equations (BDSDES) and the related stochastic partial differential equations (Zakai equations) are considered. First order algorithms are constructed using a generalized Itô-Taylor formula for two-sided stochastic differentials. The convergence order is proved through rigorous error analysis. Numerical experiments are carried out to verify the theoretical results and to demonstrate the efficiency of the proposed numerical algorithms.
Citation: Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297
References:
[1]

A. Bachouch, M. A. Ben Lasmar, A. Matoussi and M. Mnif, Numerical scheme for semilinear stochastic pdes via backward doubly stochastic differential equations, arXiv:1302.0440.

[2]

V. Bally, Approximation scheme for solutions of BSDE, in Backward stochastic differential equations (Paris, 1995-1996), vol. 364 of Pitman Res. Notes Math. Ser., Longman, Harlow, (1997), 177-191.

[3]

V. Bally and A. Matoussi, Weak solutions for SPDEs and backward doubly stochastic differential equations, J. Theoret. Probab., 14 (2001), 125-164. doi: 10.1023/A:1007825232513.

[4]

F. Bao, Y. Cao and W. Zhao, Numerical solutions for forward backward doubly stochastic differential equations and zakai equations, International Journal for Uncertainty Quantification, 1 (2011), 351-367. doi: 10.1615/Int.J.UncertaintyQuantification.2011003508.

[5]

A. Bensoussan, R. Glowinski and A. Răşcanu, Approximation of some stochastic differential equations by the splitting up method, Appl. Math. Optim., 25 (1992), 81-106. doi: 10.1007/BF01184157.

[6]

A. Budhiraja and G. Kallianpur, Approximations to the solution of the Zakai equation using multiple Wiener and Stratonovich integral expansions, Stochastics Stochastics Rep., 56 (1996), 271-315. doi: 10.1080/17442509608834046.

[7]

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

[8]

A. Davie and J. Gaines, Convergence of numerical schemes for the solution of the parabolic stochastic partial differential equations, Math. Comp., 70 (2001), 121-134. doi: 10.1090/S0025-5718-00-01224-2.

[9]

E. Gobet, G. Pagès, H. Pham and J. Printems, Discretization and simulation of the Zakai equation, SIAM J. Numer. Anal., 44 (2006), 2505-2538 (electronic). doi: 10.1137/050623140.

[10]

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.

[11]

I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II, Potential Anal., 11 (1999), 1-37. doi: 10.1023/A:1008699504438.

[12]

I. Gyöngy and N. Krylov, On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31 (2003), 564-591. doi: 10.1214/aop/1048516528.

[13]

I. Gyöngy and D. Nualart, Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise, Stochastic Process. Appl., 58 (1995), 57-72. doi: 10.1016/0304-4149(95)00010-5.

[14]

Y. Han, S. Peng and Z. Wu, Maximum principle for backward doubly stochastic control systems with applications, SIAM J. Control Optim., 48 (2010), 4224-4241. doi: 10.1137/080743561.

[15]

Y. Hu, G. Kallianpur and J. Xiong, An approximation for zakai equation, Appl. Math. Optim., 45 (2002), 23-44. doi: 10.1007/s00245-001-0024-8.

[16]

S. Janković, J. Djordjević and M. Jovanović, On a class of backward doubly stochastic differential equations, Appl. Math. Comput., 217 (2011), 8754-8764. doi: 10.1016/j.amc.2011.03.128.

[17]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[18]

J. Ma and J. Yong, Approximate solvability of forward-backward stochastic differential equations, Appl. Math. Optim., 45 (2002), 1-22. doi: 10.1007/s00245-001-0025-7.

[19]

J. Ma, P. Protter, J. San Martín and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 302-316. doi: 10.1214/aoap/1015961165.

[20]

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.

[21]

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.

[22]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, vol. 1702 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.

[23]

É. Pardoux and P. Protter, A two-sided stochastic integral and its calculus, Probab. Theory Related Fields, 76 (1987), 15-49. doi: 10.1007/BF00390274.

[24]

É. Pardoux and S. G. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Related Fields, 98 (1994), 209-227. doi: 10.1007/BF01192514.

[25]

E. Platen, An introduction to numerical methods for stochastic differential equations, in Acta numerica, 1999, vol. 8 of Acta Numer., Cambridge Univ. Press, Cambridge, (1999), 197-246. doi: 10.1017/S0962492900002920.

[26]

P. Protter and D. Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab., 25 (1997), 393-423. doi: 10.1214/aop/1024404293.

[27]

A. B. Sow, Backward doubly stochastic differential equations driven by Levy process: the case of non-Liphschitz coefficients, J. Numer. Math. Stoch., 3 (2011), 71-79.

[28]

M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11 (1969), 230-243. doi: 10.1007/BF00536382.

[29]

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

[30]

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.

[31]

W. Zhao, J. Wang and S. Peng, Error estimates of the $\theta$-scheme for backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 905-924. doi: 10.3934/dcdsb.2009.12.905.

[32]

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]

A. Bachouch, M. A. Ben Lasmar, A. Matoussi and M. Mnif, Numerical scheme for semilinear stochastic pdes via backward doubly stochastic differential equations, arXiv:1302.0440.

[2]

V. Bally, Approximation scheme for solutions of BSDE, in Backward stochastic differential equations (Paris, 1995-1996), vol. 364 of Pitman Res. Notes Math. Ser., Longman, Harlow, (1997), 177-191.

[3]

V. Bally and A. Matoussi, Weak solutions for SPDEs and backward doubly stochastic differential equations, J. Theoret. Probab., 14 (2001), 125-164. doi: 10.1023/A:1007825232513.

[4]

F. Bao, Y. Cao and W. Zhao, Numerical solutions for forward backward doubly stochastic differential equations and zakai equations, International Journal for Uncertainty Quantification, 1 (2011), 351-367. doi: 10.1615/Int.J.UncertaintyQuantification.2011003508.

[5]

A. Bensoussan, R. Glowinski and A. Răşcanu, Approximation of some stochastic differential equations by the splitting up method, Appl. Math. Optim., 25 (1992), 81-106. doi: 10.1007/BF01184157.

[6]

A. Budhiraja and G. Kallianpur, Approximations to the solution of the Zakai equation using multiple Wiener and Stratonovich integral expansions, Stochastics Stochastics Rep., 56 (1996), 271-315. doi: 10.1080/17442509608834046.

[7]

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

[8]

A. Davie and J. Gaines, Convergence of numerical schemes for the solution of the parabolic stochastic partial differential equations, Math. Comp., 70 (2001), 121-134. doi: 10.1090/S0025-5718-00-01224-2.

[9]

E. Gobet, G. Pagès, H. Pham and J. Printems, Discretization and simulation of the Zakai equation, SIAM J. Numer. Anal., 44 (2006), 2505-2538 (electronic). doi: 10.1137/050623140.

[10]

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.

[11]

I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II, Potential Anal., 11 (1999), 1-37. doi: 10.1023/A:1008699504438.

[12]

I. Gyöngy and N. Krylov, On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31 (2003), 564-591. doi: 10.1214/aop/1048516528.

[13]

I. Gyöngy and D. Nualart, Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise, Stochastic Process. Appl., 58 (1995), 57-72. doi: 10.1016/0304-4149(95)00010-5.

[14]

Y. Han, S. Peng and Z. Wu, Maximum principle for backward doubly stochastic control systems with applications, SIAM J. Control Optim., 48 (2010), 4224-4241. doi: 10.1137/080743561.

[15]

Y. Hu, G. Kallianpur and J. Xiong, An approximation for zakai equation, Appl. Math. Optim., 45 (2002), 23-44. doi: 10.1007/s00245-001-0024-8.

[16]

S. Janković, J. Djordjević and M. Jovanović, On a class of backward doubly stochastic differential equations, Appl. Math. Comput., 217 (2011), 8754-8764. doi: 10.1016/j.amc.2011.03.128.

[17]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[18]

J. Ma and J. Yong, Approximate solvability of forward-backward stochastic differential equations, Appl. Math. Optim., 45 (2002), 1-22. doi: 10.1007/s00245-001-0025-7.

[19]

J. Ma, P. Protter, J. San Martín and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12 (2002), 302-316. doi: 10.1214/aoap/1015961165.

[20]

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.

[21]

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.

[22]

J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and Their Applications, vol. 1702 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.

[23]

É. Pardoux and P. Protter, A two-sided stochastic integral and its calculus, Probab. Theory Related Fields, 76 (1987), 15-49. doi: 10.1007/BF00390274.

[24]

É. Pardoux and S. G. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Related Fields, 98 (1994), 209-227. doi: 10.1007/BF01192514.

[25]

E. Platen, An introduction to numerical methods for stochastic differential equations, in Acta numerica, 1999, vol. 8 of Acta Numer., Cambridge Univ. Press, Cambridge, (1999), 197-246. doi: 10.1017/S0962492900002920.

[26]

P. Protter and D. Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab., 25 (1997), 393-423. doi: 10.1214/aop/1024404293.

[27]

A. B. Sow, Backward doubly stochastic differential equations driven by Levy process: the case of non-Liphschitz coefficients, J. Numer. Math. Stoch., 3 (2011), 71-79.

[28]

M. Zakai, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 11 (1969), 230-243. doi: 10.1007/BF00536382.

[29]

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

[30]

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.

[31]

W. Zhao, J. Wang and S. Peng, Error estimates of the $\theta$-scheme for backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 905-924. doi: 10.3934/dcdsb.2009.12.905.

[32]

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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