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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

Feng Bao 1, , Yanzhao Cao 2, and  Weidong Zhao 3,

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.
Keywords: backward SDEs, SPDEs, Zakai equation, conditional expectation., SPDEs.
Mathematics Subject Classification: Primary: 52B10, 65D18, 68U05; Secondary: 68U0.
Citation: Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete & 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,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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