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On the Alekseev-Gröbner formula in Banach spaces
Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China |
This paper first establishes a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations with non-globally Lipschitz coefficients. Then two novel explicit schemes are designed and their convergence rates are exactly identified via the fundamental theorem. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate appropriate assumptions to allow for its super-linear growth. However, we require that the Lévy measure is finite. New arguments are developed to handle essential difficulties in the convergence analysis, caused by the super-linear growth of the jump coefficient and the fact that higher moment bounds of the Poisson increments $ \int_t^{t+h} \int_Z \,\bar{N}(\mbox{d}s,\mbox{d}z), t \geq 0, h >0 $ contribute to magnitude not more than $ O(h) $. Numerical results are finally reported to confirm the theoretical findings.
References:
[1] |
A. Andersson and R. Kruse,
Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition, BIT Numer. Math., 57 (2017), 21-53.
doi: 10.1007/s10543-016-0624-y. |
[2] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009.
doi: 10.1017/CBO9780511809781.![]() ![]() ![]() |
[3] |
W.-J. Beyn, E. Isaak and R. Kruse,
Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes, J. Sci. Comput., 67 (2016), 955-987.
doi: 10.1007/s10915-015-0114-4. |
[4] |
W.-J. Beyn, E. Isaak and R. Kruse,
Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, J. Sci. Comput., 70 (2017), 1042-1077.
doi: 10.1007/s10915-016-0290-x. |
[5] |
N. Bruti-Liberati and E. Platen,
Strong approximations of stochastic differential equations with jumps, J. Comput. Appl. Math., 205 (2007), 982-1001.
doi: 10.1016/j.cam.2006.03.040. |
[6] |
K. Dareiotis, C. Kumar and S. Sabanis,
On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations, SIAM J. Numer. Anal., 54 (2016), 1840-1872.
doi: 10.1137/151004872. |
[7] |
S. Deng, W. Fei, W. Liu and X. Mao,
The truncated EM method for stochastic differential equations with Poisson jumps, J. Comput. Appl. Math., 355 (2019), 232-257.
doi: 10.1016/j.cam.2019.01.020. |
[8] |
W. Fang and M. B. Giles, Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift: Part Ⅰ, finite time interval, preprint, arXiv: 1609.08101. |
[9] |
A. Gardoń,
The order of approximation for solutions of Itô-type stochastic differential equations with jumps, Stoch. Anal. Appl., 22 (2004), 679-699.
doi: 10.1081/SAP-120030451. |
[10] |
I. Gyöngy and N. V. Krylov,
On stochastic equations with respect to semimartingales Ⅰ, Stoch., 4 (1980), 1-21.
doi: 10.1080/03610918008833154. |
[11] |
D. J. Higham and P. E. Kloeden,
Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), 101-119.
doi: 10.1007/s00211-005-0611-8. |
[12] |
D. J. Higham, X. Mao and A. M. Stuart,
Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.
doi: 10.1137/S0036142901389530. |
[13] |
D. J. Higham and P. E. Kloeden,
Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956.
doi: 10.1016/j.cam.2006.03.039. |
[14] |
L. Hu and S. Gan,
Convergence and stability of the balanced methods for stochastic differential equations with jumps, Int. J. Comput. Math., 88 (2011), 2089-2108.
doi: 10.1080/00207160.2010.521548. |
[15] |
M. Hutzenthaler and A. Jentzen,
Convergence of the stochastic Euler scheme for locally Lipschitz coefficients, Found. Comput. Math., 11 (2011), 657-706.
doi: 10.1007/s10208-011-9101-9. |
[16] |
M. Hutzenthaler and A. Jentzen, Numerical Approximations of Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients, American Mathematical Society, 2015.
doi: 10.1090/memo/1112. |
[17] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden,
Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. A, 467 (2011), 1563-1576.
doi: 10.1098/rspa.2010.0348. |
[18] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden,
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.
doi: 10.1214/11-AAP803. |
[19] |
M. Hutzenthaler and A. Jentzen, On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients, preprint, arXiv: 1401.0295. |
[20] |
M. Hutzenthaler, A. Jentzen and X. Wang,
Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations, Math. Comp., 87 (2018), 1353-1413.
doi: 10.1090/mcom/3146. |
[21] |
J. Jacod, T. G. Kurtz, S. Méléard and P. Protter,
The approximate Euler method for Lévy driven stochastic differential equations, Ann. Inst. H. Poincaré–PR, 41 (2005), 523-558.
doi: 10.1016/j.anihpb.2004.01.007. |
[22] |
C. Kelly and G. J. Lord,
Adaptive time-stepping strategies for nonlinear stochastic systems, IMA J. Numer. Anal., 38 (2018), 1523-1549.
doi: 10.1093/imanum/drx036. |
[23] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.
doi: 10.1007/978-3-662-12616-5. |
[24] |
A. Kohatsu-Higa and P. Tankov,
Jump-adapted discretization schemes for Lévy-driven SDEs, Stoch. Proc. Appl., 120 (2010), 2258-2285.
doi: 10.1016/j.spa.2010.07.001. |
[25] |
C. Kumar and S. Sabanis, On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients, Electron. J. Probab., 22 (2017), No. 73, 1–19.
doi: 10.1214/17-EJP89. |
[26] |
C. Kumar and S. Sabanis,
On tamed Milstein schemes of SDEs driven by Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 421-463.
doi: 10.3934/dcdsb.2017020. |
[27] |
W. Liu and X. Mao,
Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, Appl. Math. Comput., 223 (2013), 389-400.
doi: 10.1016/j.amc.2013.08.023. |
[28] |
X. Q. Liu and C. W. Li,
Weak approximations and extrapolations of stochastic differential equations with jumps, SIAM J. Numer. Anal, 37 (2000), 1747-1767.
doi: 10.1137/S0036142998344512. |
[29] |
Y. Maghsoodi, Mean-square efficient numerical solution of jump-diffusion stochastic differential equations, Sankhy$\bar{a}$ Ser. A., 58 (1996), 25–47. Available from: https://www.jstor.org/stable/25051081. |
[30] |
X. Mao and L. Szpruch,
Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 238 (2013), 14-28.
doi: 10.1016/j.cam.2012.08.015. |
[31] |
X. Mao and L. Szpruch,
Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stoch., 85 (2013), 144-171.
doi: 10.1080/17442508.2011.651213. |
[32] |
X. Mao,
The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370-384.
doi: 10.1016/j.cam.2015.06.002. |
[33] |
X. Mao,
Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 296 (2016), 362-375.
doi: 10.1016/j.cam.2015.09.035. |
[34] |
R. Mikulevicius and H. Pragarauskas,
On ${L}_{p}$-estimates of some singular integrals related to jump processes, SIAM J. Math. Anal., 44 (2012), 2305-2328.
doi: 10.1137/110844854. |
[35] |
G. N. Milstein,
A theorem on the order of convergence of mean-square approximations of solutions of systems of stochastic differential equations, Tero. Prob. Appl., 32 (1987), 809-811.
doi: 10.1137/1132113. |
[36] |
G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, 2004.
doi: 10.1007/978-3-662-10063-9. |
[37] |
G. N. Milstein and M. V. Tretyakov,
Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients, SIAM J. Numer. Anal., 43 (2005), 1139-1154.
doi: 10.1137/040612026. |
[38] |
E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-13694-8. |
[39] |
P. Protter, Stochastic Integration and Differential Equations: A New Approach, Springer, Berlin, 1990.
doi: 10.1007/978-3-662-02619-9. |
[40] |
S. Sabanis,
Euler approximations with varying coefficients: the case of super-linearly growing diffusion coefficients, Ann. Appl. Probab., 26 (2016), 2083-2105.
doi: 10.1214/15-AAP1140. |
[41] |
S. Sabanis,
A note on tamed Euler approximations, Electron. Commun. Probab, 18 (2013), 1-10.
doi: 10.1214/ECP.v18-2824. |
[42] |
S. Sabanis and Y. Zhang,
On explicit order 1.5 approximations with varying coefficients: the case of super-linear diffusion coefficients, J. Complexity, 50 (2019), 84-115.
doi: 10.1016/j.jco.2018.09.004. |
[43] |
Ł Szpruch and X. Zhāng,
$V$-integrability, asymptotic stability and comparison property of explicit numerical schemes for non-linear SDEs, Math. Comp., 87 (2018), 755-783.
doi: 10.1090/mcom/3219. |
[44] |
A. Tambue and J. D. Mukam, Strong convergence of the tamed and the semi-tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz condition, preprint, arXiv: 1510.04729. |
[45] |
M. V. Tretyakov and Z. Zhang,
A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal., 51 (2013), 3135-3162.
doi: 10.1137/120902318. |
[46] |
X. Wang and S. Gan,
Compensated stochastic theta methods for stochastic differential equations with jumps, Appl. Numer. Math., 60 (2010), 877-887.
doi: 10.1016/j.apnum.2010.04.012. |
[47] |
X. Wang and S. Gan,
The tamed milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Differ. Equ. Appl., 19 (2013), 466-490.
doi: 10.1080/10236198.2012.656617. |
[48] |
X. Yang and X. Wang,
A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models, Numer. Algor., 74 (2017), 39-57.
doi: 10.1007/s11075-016-0137-4. |
[49] |
Z. Zhang and H. Ma,
Order-preserving strong schemes for SDEs with locally Lipschitz coefficients, Appl. Numer. Math., 112 (2017), 1-16.
doi: 10.1016/j.apnum.2016.09.013. |
[50] |
Z. Zhang, New explicit balanced schemes for SDEs with locally Lipschitz coefficients, preprint, arXiv: 1402.3708. |
[51] |
X. Zong, F. Wu and C. Huang,
Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients, Appl. Math. Comput., 228 (2014), 240-250.
doi: 10.1016/j.amc.2013.11.100. |
show all references
References:
[1] |
A. Andersson and R. Kruse,
Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition, BIT Numer. Math., 57 (2017), 21-53.
doi: 10.1007/s10543-016-0624-y. |
[2] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009.
doi: 10.1017/CBO9780511809781.![]() ![]() ![]() |
[3] |
W.-J. Beyn, E. Isaak and R. Kruse,
Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes, J. Sci. Comput., 67 (2016), 955-987.
doi: 10.1007/s10915-015-0114-4. |
[4] |
W.-J. Beyn, E. Isaak and R. Kruse,
Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, J. Sci. Comput., 70 (2017), 1042-1077.
doi: 10.1007/s10915-016-0290-x. |
[5] |
N. Bruti-Liberati and E. Platen,
Strong approximations of stochastic differential equations with jumps, J. Comput. Appl. Math., 205 (2007), 982-1001.
doi: 10.1016/j.cam.2006.03.040. |
[6] |
K. Dareiotis, C. Kumar and S. Sabanis,
On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations, SIAM J. Numer. Anal., 54 (2016), 1840-1872.
doi: 10.1137/151004872. |
[7] |
S. Deng, W. Fei, W. Liu and X. Mao,
The truncated EM method for stochastic differential equations with Poisson jumps, J. Comput. Appl. Math., 355 (2019), 232-257.
doi: 10.1016/j.cam.2019.01.020. |
[8] |
W. Fang and M. B. Giles, Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift: Part Ⅰ, finite time interval, preprint, arXiv: 1609.08101. |
[9] |
A. Gardoń,
The order of approximation for solutions of Itô-type stochastic differential equations with jumps, Stoch. Anal. Appl., 22 (2004), 679-699.
doi: 10.1081/SAP-120030451. |
[10] |
I. Gyöngy and N. V. Krylov,
On stochastic equations with respect to semimartingales Ⅰ, Stoch., 4 (1980), 1-21.
doi: 10.1080/03610918008833154. |
[11] |
D. J. Higham and P. E. Kloeden,
Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005), 101-119.
doi: 10.1007/s00211-005-0611-8. |
[12] |
D. J. Higham, X. Mao and A. M. Stuart,
Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.
doi: 10.1137/S0036142901389530. |
[13] |
D. J. Higham and P. E. Kloeden,
Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956.
doi: 10.1016/j.cam.2006.03.039. |
[14] |
L. Hu and S. Gan,
Convergence and stability of the balanced methods for stochastic differential equations with jumps, Int. J. Comput. Math., 88 (2011), 2089-2108.
doi: 10.1080/00207160.2010.521548. |
[15] |
M. Hutzenthaler and A. Jentzen,
Convergence of the stochastic Euler scheme for locally Lipschitz coefficients, Found. Comput. Math., 11 (2011), 657-706.
doi: 10.1007/s10208-011-9101-9. |
[16] |
M. Hutzenthaler and A. Jentzen, Numerical Approximations of Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients, American Mathematical Society, 2015.
doi: 10.1090/memo/1112. |
[17] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden,
Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. A, 467 (2011), 1563-1576.
doi: 10.1098/rspa.2010.0348. |
[18] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden,
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.
doi: 10.1214/11-AAP803. |
[19] |
M. Hutzenthaler and A. Jentzen, On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients, preprint, arXiv: 1401.0295. |
[20] |
M. Hutzenthaler, A. Jentzen and X. Wang,
Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations, Math. Comp., 87 (2018), 1353-1413.
doi: 10.1090/mcom/3146. |
[21] |
J. Jacod, T. G. Kurtz, S. Méléard and P. Protter,
The approximate Euler method for Lévy driven stochastic differential equations, Ann. Inst. H. Poincaré–PR, 41 (2005), 523-558.
doi: 10.1016/j.anihpb.2004.01.007. |
[22] |
C. Kelly and G. J. Lord,
Adaptive time-stepping strategies for nonlinear stochastic systems, IMA J. Numer. Anal., 38 (2018), 1523-1549.
doi: 10.1093/imanum/drx036. |
[23] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.
doi: 10.1007/978-3-662-12616-5. |
[24] |
A. Kohatsu-Higa and P. Tankov,
Jump-adapted discretization schemes for Lévy-driven SDEs, Stoch. Proc. Appl., 120 (2010), 2258-2285.
doi: 10.1016/j.spa.2010.07.001. |
[25] |
C. Kumar and S. Sabanis, On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients, Electron. J. Probab., 22 (2017), No. 73, 1–19.
doi: 10.1214/17-EJP89. |
[26] |
C. Kumar and S. Sabanis,
On tamed Milstein schemes of SDEs driven by Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 421-463.
doi: 10.3934/dcdsb.2017020. |
[27] |
W. Liu and X. Mao,
Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, Appl. Math. Comput., 223 (2013), 389-400.
doi: 10.1016/j.amc.2013.08.023. |
[28] |
X. Q. Liu and C. W. Li,
Weak approximations and extrapolations of stochastic differential equations with jumps, SIAM J. Numer. Anal, 37 (2000), 1747-1767.
doi: 10.1137/S0036142998344512. |
[29] |
Y. Maghsoodi, Mean-square efficient numerical solution of jump-diffusion stochastic differential equations, Sankhy$\bar{a}$ Ser. A., 58 (1996), 25–47. Available from: https://www.jstor.org/stable/25051081. |
[30] |
X. Mao and L. Szpruch,
Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 238 (2013), 14-28.
doi: 10.1016/j.cam.2012.08.015. |
[31] |
X. Mao and L. Szpruch,
Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stoch., 85 (2013), 144-171.
doi: 10.1080/17442508.2011.651213. |
[32] |
X. Mao,
The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370-384.
doi: 10.1016/j.cam.2015.06.002. |
[33] |
X. Mao,
Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 296 (2016), 362-375.
doi: 10.1016/j.cam.2015.09.035. |
[34] |
R. Mikulevicius and H. Pragarauskas,
On ${L}_{p}$-estimates of some singular integrals related to jump processes, SIAM J. Math. Anal., 44 (2012), 2305-2328.
doi: 10.1137/110844854. |
[35] |
G. N. Milstein,
A theorem on the order of convergence of mean-square approximations of solutions of systems of stochastic differential equations, Tero. Prob. Appl., 32 (1987), 809-811.
doi: 10.1137/1132113. |
[36] |
G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, 2004.
doi: 10.1007/978-3-662-10063-9. |
[37] |
G. N. Milstein and M. V. Tretyakov,
Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients, SIAM J. Numer. Anal., 43 (2005), 1139-1154.
doi: 10.1137/040612026. |
[38] |
E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-13694-8. |
[39] |
P. Protter, Stochastic Integration and Differential Equations: A New Approach, Springer, Berlin, 1990.
doi: 10.1007/978-3-662-02619-9. |
[40] |
S. Sabanis,
Euler approximations with varying coefficients: the case of super-linearly growing diffusion coefficients, Ann. Appl. Probab., 26 (2016), 2083-2105.
doi: 10.1214/15-AAP1140. |
[41] |
S. Sabanis,
A note on tamed Euler approximations, Electron. Commun. Probab, 18 (2013), 1-10.
doi: 10.1214/ECP.v18-2824. |
[42] |
S. Sabanis and Y. Zhang,
On explicit order 1.5 approximations with varying coefficients: the case of super-linear diffusion coefficients, J. Complexity, 50 (2019), 84-115.
doi: 10.1016/j.jco.2018.09.004. |
[43] |
Ł Szpruch and X. Zhāng,
$V$-integrability, asymptotic stability and comparison property of explicit numerical schemes for non-linear SDEs, Math. Comp., 87 (2018), 755-783.
doi: 10.1090/mcom/3219. |
[44] |
A. Tambue and J. D. Mukam, Strong convergence of the tamed and the semi-tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz condition, preprint, arXiv: 1510.04729. |
[45] |
M. V. Tretyakov and Z. Zhang,
A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal., 51 (2013), 3135-3162.
doi: 10.1137/120902318. |
[46] |
X. Wang and S. Gan,
Compensated stochastic theta methods for stochastic differential equations with jumps, Appl. Numer. Math., 60 (2010), 877-887.
doi: 10.1016/j.apnum.2010.04.012. |
[47] |
X. Wang and S. Gan,
The tamed milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Differ. Equ. Appl., 19 (2013), 466-490.
doi: 10.1080/10236198.2012.656617. |
[48] |
X. Yang and X. Wang,
A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models, Numer. Algor., 74 (2017), 39-57.
doi: 10.1007/s11075-016-0137-4. |
[49] |
Z. Zhang and H. Ma,
Order-preserving strong schemes for SDEs with locally Lipschitz coefficients, Appl. Numer. Math., 112 (2017), 1-16.
doi: 10.1016/j.apnum.2016.09.013. |
[50] |
Z. Zhang, New explicit balanced schemes for SDEs with locally Lipschitz coefficients, preprint, arXiv: 1402.3708. |
[51] |
X. Zong, F. Wu and C. Huang,
Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients, Appl. Math. Comput., 228 (2014), 240-250.
doi: 10.1016/j.amc.2013.11.100. |

$h$ | CPU time (second) | |||
non-additive case | additive case | |||
tamed method | sine method | tamed method | sine method | |
$2^{-8}$ | 0.869440 | 0.744448 | 0.794577 | 0.632519 |
$2^{-9}$ | 1.203300 | 0.932720 | 1.031226 | 0.776562 |
$2^{-10}$ | 1.625105 | 1.100245 | 1.276387 | 0.906966 |
$2^{-11}$ | 2.951017 | 1.887072 | 2.355608 | 1.433305 |
$2^{-12}$ | 5.789335 | 3.588197 | 4.325145 | 2.473830 |
$h$ | CPU time (second) | |||
non-additive case | additive case | |||
tamed method | sine method | tamed method | sine method | |
$2^{-8}$ | 0.869440 | 0.744448 | 0.794577 | 0.632519 |
$2^{-9}$ | 1.203300 | 0.932720 | 1.031226 | 0.776562 |
$2^{-10}$ | 1.625105 | 1.100245 | 1.276387 | 0.906966 |
$2^{-11}$ | 2.951017 | 1.887072 | 2.355608 | 1.433305 |
$2^{-12}$ | 5.789335 | 3.588197 | 4.325145 | 2.473830 |
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