# American Institute of Mathematical Sciences

August  2019, 24(8): 4513-4545. doi: 10.3934/dcdsb.2019154

## 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

* Corresponding author: x.j.wang7@csu.edu.cn; x.j.wang7@gmail.com(Xiaojie Wang)

Received  November 2017 Revised  March 2019 Published  August 2019 Early access  June 2019

Fund Project: This work was supported by NSF of China (11571373, 11671405, 91630312), NSF of Hunan Province (2016JJ3137), Innovation-Driven Project of CSU (2017CX017), Shenghua Yuying Program of CSU and Hunan Provincial Innovation Foundation For Postgraduate (CX2018B051).

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.

Citation: Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4513-4545. doi: 10.3934/dcdsb.2019154
##### 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.
Mean-square convergence rates for (87) (left) and (88) (right)
CPU time of the tamed and sine methods with different stepsizes
 $h$ 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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