On tamed milstein schemes of SDEs driven by Lévy noise

Chaman Kumar; Sotirios Sabanis

doi:10.3934/dcdsb.2017020

Article Contents

2017, Volume 22, Issue 2: 421-463. Doi: 10.3934/dcdsb.2017020

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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
* Corresponding author: Sotirios Sabanis

Received: May 31, 2015

Revised: November 30, 2015

Published: December 2017

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

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Abstract

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.

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

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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)

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Table 1. Tamed Milstein scheme (60) of SDE (59)

$h=2^{-n}$	$E\|x_T-x_T^h\|$	$\sqrt{E\|x_T-x_T^h\|^2}$	$\sqrt[3]{E\|x_T-x_T^h\|^3}$	$\sqrt[4]{E\|x_T-x_T^h\|^4}$	$\sqrt[5]{E\|x_T-x_T^h\|^5}$
$2^{-20}$	0.0000007223	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

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

$h=2^{-n}$	$\sqrt{E\|x_T-x_T^h\|^2}$
$h=2^{-n}$	$\lambda=3.0$	$\lambda=5.0$
$2^{-20}$	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 $0.125$.
$h=2^{-n}$	$\sqrt{E\|x_T-x_T^h\|^2}$
$h=2^{-n}$	$\lambda=3.0$	$\lambda=5.0$
$2^{-20}$	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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