
-
Previous Article
Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals
- DCDS-B Home
- This Issue
-
Next Article
Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions
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 |
We extend the taming techniques developed in [
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. |
[2] |
R. Cont and P. Tankov,
Financial Modelling with Jump Processes, Chapman and Hall/CRC, Florida, USA, 2004. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
P. E. Kloeden and E. Platen,
Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Springer, Berlin, 1992. |
[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. |
[16] |
B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd edition, Springer, Berlin, 2007. |
[17] |
E. Platen and N. Bruti-Liberati,
Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer-Verlag, Berlin, 2010. |
[18] |
D. Revuz and M. Yor,
Continuous martingales and Brownian motion, 3rd edition, Springer-Verlag, Berlin, 1999. |
[19] |
S. Sabanis,
A note on tamed Euler approximations, Electronic Communications in Probability, 18 (2013), 1-10.
doi: 10.1214/ECP.v18-2824. |
[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. |
[21] |
R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. |
[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. |
[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. |
[24] |
Z. Zhang, New explicit balanced schemes for SDEs with locally Lipschitz coefficients, 2014, arXiv: 1402.3708. |
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. |
[2] |
R. Cont and P. Tankov,
Financial Modelling with Jump Processes, Chapman and Hall/CRC, Florida, USA, 2004. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
P. E. Kloeden and E. Platen,
Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Springer, Berlin, 1992. |
[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. |
[16] |
B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd edition, Springer, Berlin, 2007. |
[17] |
E. Platen and N. Bruti-Liberati,
Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer-Verlag, Berlin, 2010. |
[18] |
D. Revuz and M. Yor,
Continuous martingales and Brownian motion, 3rd edition, Springer-Verlag, Berlin, 1999. |
[19] |
S. Sabanis,
A note on tamed Euler approximations, Electronic Communications in Probability, 18 (2013), 1-10.
doi: 10.1214/ECP.v18-2824. |
[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. |
[21] |
R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. |
[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. |
[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. |
[24] |
Z. Zhang, New explicit balanced schemes for SDEs with locally Lipschitz coefficients, 2014, arXiv: 1402.3708. |


0.0000007223 | 0.0000027396 | 0.0000071338 | 0.0000133872 | 0.0000206505 | |
0.0000020488 | 0.0000081546 | 0.0000213011 | 0.0000400356 | 0.0000618759 | |
0.0000046829 | 0.0000190372 | 0.0000498670 | 0.0000937473 | 0.0001448460 | |
0.0000099526 | 0.0000408359 | 0.0001072262 | 0.0002022094 | 0.0003133946 | |
0.0000205589 | 0.0000844630 | 0.0002227601 | 0.0004218232 | 0.0006555927 | |
0.0000417394 | 0.0001723833 | 0.0004554010 | 0.0008642163 | 0.0013460486 | |
0.0000843948 | 0.0003519474 | 0.0009360518 | 0.0017870537 | 0.0027962853 | |
0.0001710052 | 0.0007232200 | 0.0019649384 | 0.0038259755 | 0.0060684654 | |
0.0003479789 | 0.0015293072 | 0.0043796657 | 0.0089031484 | 0.0144907359 | |
0.0007231189 | 0.0035802581 | 0.0118774764 | 0.0259649292 | 0.0432914310 |
0.0000007223 | 0.0000027396 | 0.0000071338 | 0.0000133872 | 0.0000206505 | |
0.0000020488 | 0.0000081546 | 0.0000213011 | 0.0000400356 | 0.0000618759 | |
0.0000046829 | 0.0000190372 | 0.0000498670 | 0.0000937473 | 0.0001448460 | |
0.0000099526 | 0.0000408359 | 0.0001072262 | 0.0002022094 | 0.0003133946 | |
0.0000205589 | 0.0000844630 | 0.0002227601 | 0.0004218232 | 0.0006555927 | |
0.0000417394 | 0.0001723833 | 0.0004554010 | 0.0008642163 | 0.0013460486 | |
0.0000843948 | 0.0003519474 | 0.0009360518 | 0.0017870537 | 0.0027962853 | |
0.0001710052 | 0.0007232200 | 0.0019649384 | 0.0038259755 | 0.0060684654 | |
0.0003479789 | 0.0015293072 | 0.0043796657 | 0.0089031484 | 0.0144907359 | |
0.0007231189 | 0.0035802581 | 0.0118774764 | 0.0259649292 | 0.0432914310 |
0.00067484 | 0.00621555 | |
0.00204889 | 0.02203719 | |
0.00515874 | 0.06003098 | |
0.01321011 | 0.27830910 | |
0.03146860 | 0.45542612 | |
0.06349005 | 0.66561201 | |
0.16459365 | 0.85082102 | |
0.27426757 | 1.55620505 | |
0.40437133 | 2.07850380 | |
0.57251083 | 2.41922833 | |
(A) Mark is normal with mean | ||
0.00004365 | 0.00005260 | |
0.00011286 | 0.00013058 | |
0.00025133 | 0.00028420 | |
0.00054134 | 0.00059787 | |
0.00112643 | 0.00125615 | |
0.00242178 | 0.00272857 | |
0.00563126 | 0.00673629 | |
0.01497166 | 0.01747566 | |
0.03745749 | 0.03448135 | |
0.07302734 | 0.07900926 | |
(B) Mark is uniform on |
0.00067484 | 0.00621555 | |
0.00204889 | 0.02203719 | |
0.00515874 | 0.06003098 | |
0.01321011 | 0.27830910 | |
0.03146860 | 0.45542612 | |
0.06349005 | 0.66561201 | |
0.16459365 | 0.85082102 | |
0.27426757 | 1.55620505 | |
0.40437133 | 2.07850380 | |
0.57251083 | 2.41922833 | |
(A) Mark is normal with mean | ||
0.00004365 | 0.00005260 | |
0.00011286 | 0.00013058 | |
0.00025133 | 0.00028420 | |
0.00054134 | 0.00059787 | |
0.00112643 | 0.00125615 | |
0.00242178 | 0.00272857 | |
0.00563126 | 0.00673629 | |
0.01497166 | 0.01747566 | |
0.03745749 | 0.03448135 | |
0.07302734 | 0.07900926 | |
(B) Mark is uniform on |
[1] |
Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete and 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 and 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 and Continuous Dynamical Systems - B, 2022, 27 (7) : 3663-3682. 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 and Continuous Dynamical Systems - B, 2022, 27 (7) : 3725-3747. doi: 10.3934/dcdsb.2021204 |
[5] |
Yulin Song. Density functions of distribution dependent SDEs driven by Lévy noises. Communications on Pure and 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 and 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 and 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 and 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 and 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 and Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355 |
[12] |
Yanqiang Chang, Huabin Chen. Stability analysis of stochastic delay differential equations with Markovian switching driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021301 |
[13] |
Yong Ren, Qi Zhang. Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3811-3829. doi: 10.3934/dcdsb.2021207 |
[14] |
Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and Optimization, 2012, 2 (4) : 749-765. doi: 10.3934/naco.2012.2.749 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]