Explicit numerical schemes of SDEs driven by Lévy Noise with super-linear coeffcients and their application to delay equations
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Date
26/11/2015Author
Kumar, Chaman
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Abstract
We investigate an explicit tamed Euler scheme of stochastic differential equation with
random coefficients driven by Lévy noise, which has super-linear drift coefficient. The
strong convergence property of the tamed Euler scheme is proved when drift coefficient
satisfies one-sided local Lipschitz condition whereas diffusion and jump coefficients
satisfy local Lipschitz conditions. A rate of convergence for the tamed Euler scheme is
recovered when local Lipschitz conditions are replaced by global Lipschitz conditions
and drift satisfies polynomial Lipschitz condition. These findings are consistent with
those of the classical Euler scheme. New methodologies are developed to overcome
challenges arising due to the jumps and the randomness of the coefficients. Moreover,
as an application of these findings, a tamed Euler scheme is proposed for the stochastic
delay differential equation driven by Lévy noise with drift coefficient that grows super-linearly
in both delay and non-delay variables. The strong convergence property of
the tamed Euler scheme for such SDDE driven by Lévy noise is studied and rate of
convergence is shown to be consistent with that of the classical Euler scheme. Finally,
an explicit tamed Milstein scheme with rate of convergence arbitrarily close to one
is developed to approximate the stochastic differential equation driven by Lévy noise
(without random coefficients) that has super-linearly growing drift coefficient.