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Impact of $\alpha$-stable Lévy noise on the Stommel model for the thermohaline circulation
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 |
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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