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Special section on differential equations: Theory, application, and numerical approximation
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 |
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,, , ().
|
[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,, , ().
|
[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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