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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,,
Department of Mathematics, Indian Institute of Technology, Roorkee, India
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
Figure(2) / Table(2)
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 & Continuous Dynamical Systems - B, 2017, 22 (2) : 421-463. doi: 10.3934/dcdsb.2017020
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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)
Table 1. Tamed Milstein scheme (60) of SDE (59)
$h=2^{-n}$ $E|x_T-x_T^h|$ $\sqrt{E|x_T-x_T^h|^2}$ $\sqrt[3]{E|x_T-x_T^h|^3}$ $\sqrt[4]{E|x_T-x_T^h|^4}$ $\sqrt[5]{E|x_T-x_T^h|^5}$
$2^{-20}$ 0.0000007223 0.0000027396 0.0000071338 0.0000133872 0.0000206505
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$h=2^{-n}$ $\sqrt{E|x_T-x_T^h|^2}$
$\lambda=3.0$ $\lambda=5.0$
$2^{-20}$ 0.00067484 0.00621555
(A) Mark is normal with mean $0$ and variance $0.125$.
(B) Mark is uniform on $[-1/4,1/4]$.
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Chaman Kumar Sotirios Sabanis