Article Contents
Article Contents

# On tamed milstein schemes of SDEs driven by Lévy noise

• * Corresponding author: Sotirios Sabanis
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.

Mathematics Subject Classification: Primary:60H35;secondary:65C30.

 Citation:

• Figure 1.  $\mathcal{L}^q$-convergence rate of tamed Milstein scheme (60) of SDE (59)

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 $2^{-19}$ 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

Table 2.  Tamed Milstein scheme (62) of SDE (61)

 $h=2^{-n}$ $\sqrt{E|x_T-x_T^h|^2}$ $\lambda=3.0$ $\lambda=5.0$ $2^{-20}$ 0.00067484 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 $[-1/4,1/4]$.
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