# American Institute of Mathematical Sciences

March  2017, 22(2): 421-463. doi: 10.3934/dcdsb.2017020

## On tamed milstein schemes of SDEs driven by Lévy noise

 1 Department of Mathematics, Indian Institute of Technology, Roorkee, India 2 School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom

* Corresponding author: Sotirios Sabanis

Received  June 2015 Revised  December 2015 Published  December 2016

Fund Project: This work was done when the first author was a PhD student in the School of Mathematics, University of Edinburgh, United Kingdom.

We extend the taming techniques developed in [3,19] to construct explicit Milstein schemes that numerically approximate Lévy driven stochastic differential equations with super-linearly growing drift coefficients. The classical rate of convergence is recovered when the first derivative of the drift coefficient satisfies a polynomial Lipschitz condition.

Citation: Chaman Kumar, Sotirios Sabanis. On tamed milstein schemes of SDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 421-463. doi: 10.3934/dcdsb.2017020
##### References:
 [1] N. Bruti-Liberati and E. Platen, Strong approximations of stochastic differential equations with jumps, Journal of Computational and Applied Mathematics, 205 (2007), 982-1001.  doi: 10.1016/j.cam.2006.03.040.  Google Scholar [2] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, Florida, USA, 2004. Google Scholar [3] 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.  Google Scholar [4] S. Dereich, Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction, Annals of Applied Probability, 21 (2011), 283-311.  doi: 10.1214/10-AAP695.  Google Scholar [5] S. Dereich and F. Heidenreich, A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations, Stochastic Processes and their Applications, 121 (2011), 1565-1587.  doi: 10.1016/j.spa.2011.03.015.  Google Scholar [6] I. Gyöngy and N. V. Krylov, On Stochastic Equations with Respect to Semi-martingales Ⅰ, Stochastics, 4 (1980), 1-21.  doi: 10.1080/03610918008833154.  Google Scholar [7] D. J. Higham and P. E. Kloeden, Numerical methods for non-linear stochastic differential equations with jumps, Numerische Mathematik, 110 (2005), 101-119.  doi: 10.1007/s00211-005-0611-8.  Google Scholar [8] D. J. Higham and P. E. Kloeden, Convergence and stability of implicit methods for jump-diffusion systems, International Journal of Numerical Analysis and Modelling, 3 (2006), 125-140.   Google Scholar [9] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Memoirs of the American Mathematical Society, 236 (2015), ⅴ+99 pp. Google Scholar [10] 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, 2014, arXiv: 1401.0295. Google Scholar [11] M. Hutzethaler, 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, Proceedings of the Royal Society A, 467 (2010), 1563-1576.  doi: 10.1098/rspa.2010.0348.  Google Scholar [12] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, The Annals of Applied Probability, 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803.  Google Scholar [13] J. Jacod, T. G. Kurtz, S. Méléard and P. Protter, The approximate Euler method for Lévy driven stochastic differential equations, Ann. I. H. Poincaré-PR, 41 (2005), 523-558.  doi: 10.1016/j.anihpb.2004.01.007.  Google Scholar [14] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Springer, Berlin, 1992. Google Scholar [15] R. Mikulevicius and H. Pragarauskas, On $\mathcal{L}_p$-estimates of some singular integrals related to jump processes, SIAM J. Math. Anal., 44 (2012), 2305-2328.  doi: 10.1137/110844854.  Google Scholar [16] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd edition, Springer, Berlin, 2007. Google Scholar [17] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer-Verlag, Berlin, 2010. Google Scholar [18] D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd edition, Springer-Verlag, Berlin, 1999. Google Scholar [19] S. Sabanis, A note on tamed Euler approximations, Electronic Communications in Probability, 18 (2013), 1-10.  doi: 10.1214/ECP.v18-2824.  Google Scholar [20] S. Sabanis, Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients, Ann. Appl. Probab., 26 (2016), 2083-2105.  doi: 10.1214/15-AAP1140.  Google Scholar [21] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. Google Scholar [22] M. V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM Journal of Numerical Analysis, 51 (2013), 3135-3162.  doi: 10.1137/120902318.  Google Scholar [23] X. Wang and S. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, Journal of Difference Equations and Applications, 19 (2013), 466-490.  doi: 10.1080/10236198.2012.656617.  Google Scholar [24] Z. Zhang, New explicit balanced schemes for SDEs with locally Lipschitz coefficients, 2014, arXiv: 1402.3708. Google Scholar

show all references

##### References:
 [1] N. Bruti-Liberati and E. Platen, Strong approximations of stochastic differential equations with jumps, Journal of Computational and Applied Mathematics, 205 (2007), 982-1001.  doi: 10.1016/j.cam.2006.03.040.  Google Scholar [2] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, Florida, USA, 2004. Google Scholar [3] 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.  Google Scholar [4] S. Dereich, Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction, Annals of Applied Probability, 21 (2011), 283-311.  doi: 10.1214/10-AAP695.  Google Scholar [5] S. Dereich and F. Heidenreich, A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations, Stochastic Processes and their Applications, 121 (2011), 1565-1587.  doi: 10.1016/j.spa.2011.03.015.  Google Scholar [6] I. Gyöngy and N. V. Krylov, On Stochastic Equations with Respect to Semi-martingales Ⅰ, Stochastics, 4 (1980), 1-21.  doi: 10.1080/03610918008833154.  Google Scholar [7] D. J. Higham and P. E. Kloeden, Numerical methods for non-linear stochastic differential equations with jumps, Numerische Mathematik, 110 (2005), 101-119.  doi: 10.1007/s00211-005-0611-8.  Google Scholar [8] D. J. Higham and P. E. Kloeden, Convergence and stability of implicit methods for jump-diffusion systems, International Journal of Numerical Analysis and Modelling, 3 (2006), 125-140.   Google Scholar [9] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Memoirs of the American Mathematical Society, 236 (2015), ⅴ+99 pp. Google Scholar [10] 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, 2014, arXiv: 1401.0295. Google Scholar [11] M. Hutzethaler, 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, Proceedings of the Royal Society A, 467 (2010), 1563-1576.  doi: 10.1098/rspa.2010.0348.  Google Scholar [12] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, The Annals of Applied Probability, 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803.  Google Scholar [13] J. Jacod, T. G. Kurtz, S. Méléard and P. Protter, The approximate Euler method for Lévy driven stochastic differential equations, Ann. I. H. Poincaré-PR, 41 (2005), 523-558.  doi: 10.1016/j.anihpb.2004.01.007.  Google Scholar [14] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Springer, Berlin, 1992. Google Scholar [15] R. Mikulevicius and H. Pragarauskas, On $\mathcal{L}_p$-estimates of some singular integrals related to jump processes, SIAM J. Math. Anal., 44 (2012), 2305-2328.  doi: 10.1137/110844854.  Google Scholar [16] B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd edition, Springer, Berlin, 2007. Google Scholar [17] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer-Verlag, Berlin, 2010. Google Scholar [18] D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd edition, Springer-Verlag, Berlin, 1999. Google Scholar [19] S. Sabanis, A note on tamed Euler approximations, Electronic Communications in Probability, 18 (2013), 1-10.  doi: 10.1214/ECP.v18-2824.  Google Scholar [20] S. Sabanis, Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients, Ann. Appl. Probab., 26 (2016), 2083-2105.  doi: 10.1214/15-AAP1140.  Google Scholar [21] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. Google Scholar [22] M. V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM Journal of Numerical Analysis, 51 (2013), 3135-3162.  doi: 10.1137/120902318.  Google Scholar [23] X. Wang and S. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, Journal of Difference Equations and Applications, 19 (2013), 466-490.  doi: 10.1080/10236198.2012.656617.  Google Scholar [24] Z. Zhang, New explicit balanced schemes for SDEs with locally Lipschitz coefficients, 2014, arXiv: 1402.3708. Google Scholar
$\mathcal{L}^q$-convergence rate of tamed Milstein scheme (60) of SDE (59)
$\mathcal{L}^2$-convergence rate of tamed Milstein scheme (62) of SDE (61)
Tamed Milstein scheme (60) of SDE (59)
 $h=2^{-n}$ $E|x_T-x_T^h|$ $\sqrt{E|x_T-x_T^h|^2}$ $\sqrt[3]{E|x_T-x_T^h|^3}$ $\sqrt[4]{E|x_T-x_T^h|^4}$ $\sqrt[5]{E|x_T-x_T^h|^5}$ $2^{-20}$ 0.0000007223 0.0000027396 0.0000071338 0.0000133872 0.0000206505 $2^{-19}$ 0.0000020488 0.0000081546 0.0000213011 0.0000400356 0.0000618759 $2^{-18}$ 0.0000046829 0.0000190372 0.0000498670 0.0000937473 0.0001448460 $2^{-17}$ 0.0000099526 0.0000408359 0.0001072262 0.0002022094 0.0003133946 $2^{-16}$ 0.0000205589 0.0000844630 0.0002227601 0.0004218232 0.0006555927 $2^{-15}$ 0.0000417394 0.0001723833 0.0004554010 0.0008642163 0.0013460486 $2^{-14}$ 0.0000843948 0.0003519474 0.0009360518 0.0017870537 0.0027962853 $2^{-13}$ 0.0001710052 0.0007232200 0.0019649384 0.0038259755 0.0060684654 $2^{-12}$ 0.0003479789 0.0015293072 0.0043796657 0.0089031484 0.0144907359 $2^{-11}$ 0.0007231189 0.0035802581 0.0118774764 0.0259649292 0.0432914310
 $h=2^{-n}$ $E|x_T-x_T^h|$ $\sqrt{E|x_T-x_T^h|^2}$ $\sqrt[3]{E|x_T-x_T^h|^3}$ $\sqrt[4]{E|x_T-x_T^h|^4}$ $\sqrt[5]{E|x_T-x_T^h|^5}$ $2^{-20}$ 0.0000007223 0.0000027396 0.0000071338 0.0000133872 0.0000206505 $2^{-19}$ 0.0000020488 0.0000081546 0.0000213011 0.0000400356 0.0000618759 $2^{-18}$ 0.0000046829 0.0000190372 0.0000498670 0.0000937473 0.0001448460 $2^{-17}$ 0.0000099526 0.0000408359 0.0001072262 0.0002022094 0.0003133946 $2^{-16}$ 0.0000205589 0.0000844630 0.0002227601 0.0004218232 0.0006555927 $2^{-15}$ 0.0000417394 0.0001723833 0.0004554010 0.0008642163 0.0013460486 $2^{-14}$ 0.0000843948 0.0003519474 0.0009360518 0.0017870537 0.0027962853 $2^{-13}$ 0.0001710052 0.0007232200 0.0019649384 0.0038259755 0.0060684654 $2^{-12}$ 0.0003479789 0.0015293072 0.0043796657 0.0089031484 0.0144907359 $2^{-11}$ 0.0007231189 0.0035802581 0.0118774764 0.0259649292 0.0432914310
Tamed Milstein scheme (62) of SDE (61)
 $h=2^{-n}$ $\sqrt{E|x_T-x_T^h|^2}$ $\lambda=3.0$ $\lambda=5.0$ $2^{-20}$ 0.00067484 0.00621555 $2^{-19}$ 0.00204889 0.02203719 $2^{-18}$ 0.00515874 0.06003098 $2^{-17}$ 0.01321011 0.27830910 $2^{-16}$ 0.03146860 0.45542612 $2^{-15}$ 0.06349005 0.66561201 $2^{-14}$ 0.16459365 0.85082102 $2^{-13}$ 0.27426757 1.55620505 $2^{-12}$ 0.40437133 2.07850380 $2^{-11}$ 0.57251083 2.41922833 (A) Mark is normal with mean $0$ and variance $0.125$. $h=2^{-n}$ $\sqrt{E|x_T-x_T^h|^2}$ $\lambda=3.0$ $\lambda=5.0$ $2^{-20}$ 0.00004365 0.00005260 $2^{-19}$ 0.00011286 0.00013058 $2^{-18}$ 0.00025133 0.00028420 $2^{-17}$ 0.00054134 0.00059787 $2^{-16}$ 0.00112643 0.00125615 $2^{-15}$ 0.00242178 0.00272857 $2^{-14}$ 0.00563126 0.00673629 $2^{-13}$ 0.01497166 0.01747566 $2^{-12}$ 0.03745749 0.03448135 $2^{-11}$ 0.07302734 0.07900926 (B) Mark is uniform on $[-1/4,1/4]$.
 $h=2^{-n}$ $\sqrt{E|x_T-x_T^h|^2}$ $\lambda=3.0$ $\lambda=5.0$ $2^{-20}$ 0.00067484 0.00621555 $2^{-19}$ 0.00204889 0.02203719 $2^{-18}$ 0.00515874 0.06003098 $2^{-17}$ 0.01321011 0.27830910 $2^{-16}$ 0.03146860 0.45542612 $2^{-15}$ 0.06349005 0.66561201 $2^{-14}$ 0.16459365 0.85082102 $2^{-13}$ 0.27426757 1.55620505 $2^{-12}$ 0.40437133 2.07850380 $2^{-11}$ 0.57251083 2.41922833 (A) Mark is normal with mean $0$ and variance $0.125$. $h=2^{-n}$ $\sqrt{E|x_T-x_T^h|^2}$ $\lambda=3.0$ $\lambda=5.0$ $2^{-20}$ 0.00004365 0.00005260 $2^{-19}$ 0.00011286 0.00013058 $2^{-18}$ 0.00025133 0.00028420 $2^{-17}$ 0.00054134 0.00059787 $2^{-16}$ 0.00112643 0.00125615 $2^{-15}$ 0.00242178 0.00272857 $2^{-14}$ 0.00563126 0.00673629 $2^{-13}$ 0.01497166 0.01747566 $2^{-12}$ 0.03745749 0.03448135 $2^{-11}$ 0.07302734 0.07900926 (B) Mark is uniform on $[-1/4,1/4]$.
 [1] Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1405-1446. doi: 10.3934/dcdsb.2020167 [2] Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4513-4545. doi: 10.3934/dcdsb.2019154 [3] Weijun Zhan, Qian Guo, Yuhao Cong. The truncated Milstein method for super-linear stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021201 [4] Tian Zhang, Chuanhou Gao. Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021204 [5] Yulin Song. Density functions of distribution dependent SDEs driven by Lévy noises. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2399-2419. doi: 10.3934/cpaa.2021087 [6] Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002 [7] Linghua Chen, Espen R. Jakobsen. L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5735-5763. doi: 10.3934/dcds.2018250 [8] E. N. Dancer. On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W-M Ni and L. Su. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3861-3869. doi: 10.3934/dcds.2012.32.3861 [9] Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282 [10] Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080 [11] Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355 [12] Yong Ren, Qi Zhang. Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021207 [13] Yanqiang Chang, Huabin Chen. Stability analysis of stochastic delay differential equations with Markovian switching driven by L${\acute{e}}$vy noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021301 [14] Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2083-2100. doi: 10.3934/dcdsb.2013.18.2083 [15] Karel Kadlec, Bohdan Maslowski. Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 4039-4055. doi: 10.3934/dcdsb.2020137 [16] Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057 [17] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [18] Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 227-245. doi: 10.3934/dcdsb.2017012 [19] Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027 [20] Rachel Chen, Jianqiang Hu, Yijie Peng. Simulation of Lévy-Driven models and its application in finance. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 749-765. doi: 10.3934/naco.2012.2.749

2020 Impact Factor: 1.327