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

On tamed milstein schemes of SDEs driven by Lévy noise

Chaman Kumar 1, and  Sotirios Sabanis 2,,

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.

Keywords: Explicit Milstein scheme, super-linear growth, rate of convergence, SDEs driven by Lévy noise.
Mathematics Subject Classification: Primary: 60H35; secondary: 65C3.
Citation: Chaman Kumar, Sotirios Sabanis. On tamed milstein schemes of SDEs driven by Lévy noise. Discrete and 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.

[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.

Figure 1.  $\mathcal{L}^q$-convergence rate of tamed Milstein scheme (60) of SDE (59)
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Figure 2.  $\mathcal{L}^2$-convergence rate of tamed Milstein scheme (62) of SDE (61)
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Table 1.  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}$$\sqrt[5]{E|x_T-x_T^h|^5}$
$2^{-20}$0.00000072230.00000273960.00000713380.00001338720.0000206505
$2^{-19}$0.00000204880.00000815460.00002130110.00004003560.0000618759
$2^{-18}$0.00000468290.00001903720.00004986700.00009374730.0001448460
$2^{-17}$0.00000995260.00004083590.00010722620.00020220940.0003133946
$2^{-16}$0.00002055890.00008446300.00022276010.00042182320.0006555927
$2^{-15}$0.00004173940.00017238330.00045540100.00086421630.0013460486
$2^{-14}$0.00008439480.00035194740.00093605180.00178705370.0027962853
$2^{-13}$0.00017100520.00072322000.00196493840.00382597550.0060684654
$2^{-12}$0.00034797890.00152930720.00437966570.00890314840.0144907359
$2^{-11}$0.00072311890.00358025810.01187747640.02596492920.0432914310
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$h=2^{-n}$$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}$0.00000072230.00000273960.00000713380.00001338720.0000206505
$2^{-19}$0.00000204880.00000815460.00002130110.00004003560.0000618759
$2^{-18}$0.00000468290.00001903720.00004986700.00009374730.0001448460
$2^{-17}$0.00000995260.00004083590.00010722620.00020220940.0003133946
$2^{-16}$0.00002055890.00008446300.00022276010.00042182320.0006555927
$2^{-15}$0.00004173940.00017238330.00045540100.00086421630.0013460486
$2^{-14}$0.00008439480.00035194740.00093605180.00178705370.0027962853
$2^{-13}$0.00017100520.00072322000.00196493840.00382597550.0060684654
$2^{-12}$0.00034797890.00152930720.00437966570.00890314840.0144907359
$2^{-11}$0.00072311890.00358025810.01187747640.02596492920.0432914310
Table 2.  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}$0.000674840.00621555
$2^{-19}$0.002048890.02203719
$2^{-18}$0.005158740.06003098
$2^{-17}$0.013210110.27830910
$2^{-16}$0.031468600.45542612
$2^{-15}$0.063490050.66561201
$2^{-14}$0.164593650.85082102
$2^{-13}$0.274267571.55620505
$2^{-12}$0.404371332.07850380
$2^{-11}$0.572510832.41922833
(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}$0.000043650.00005260
$2^{-19}$0.000112860.00013058
$2^{-18}$0.000251330.00028420
$2^{-17}$0.000541340.00059787
$2^{-16}$0.001126430.00125615
$2^{-15}$0.002421780.00272857
$2^{-14}$0.005631260.00673629
$2^{-13}$0.014971660.01747566
$2^{-12}$0.037457490.03448135
$2^{-11}$0.073027340.07900926
(B) Mark is uniform on $[-1/4,1/4]$.
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$h=2^{-n}$$\sqrt{E|x_T-x_T^h|^2}$
$\lambda=3.0$$\lambda=5.0$
$2^{-20}$0.000674840.00621555
$2^{-19}$0.002048890.02203719
$2^{-18}$0.005158740.06003098
$2^{-17}$0.013210110.27830910
$2^{-16}$0.031468600.45542612
$2^{-15}$0.063490050.66561201
$2^{-14}$0.164593650.85082102
$2^{-13}$0.274267571.55620505
$2^{-12}$0.404371332.07850380
$2^{-11}$0.572510832.41922833
(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}$0.000043650.00005260
$2^{-19}$0.000112860.00013058
$2^{-18}$0.000251330.00028420
$2^{-17}$0.000541340.00059787
$2^{-16}$0.001126430.00125615
$2^{-15}$0.002421780.00272857
$2^{-14}$0.005631260.00673629
$2^{-13}$0.014971660.01747566
$2^{-12}$0.037457490.03448135
$2^{-11}$0.073027340.07900926
(B) Mark is uniform on $[-1/4,1/4]$.
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