PRJan 22, 2015
On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay EquationsKonstantinos Dareiotis, Chaman Kumar, Sotirios Sabanis
We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by Lévy noise with super-linearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations.
PRJan 11, 2016
On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficientsChaman Kumar, Sotirios Sabanis
A new class of explicit Milstein schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, is proposed in this article. It is shown, under very mild conditions, that these explicit schemes converge in $\mathcal L^p$ to the solution of the corresponding SDEs with optimal rate.
PRNov 10, 2016
On Explicit Approximations for Lévy Driven SDEs with Super-linear Diffusion CoefficientsChaman Kumar, Sotirios Sabanis
Motivated by the results of \cite{sabanis2015}, we propose explicit Euler-type schemes for SDEs with random coefficients driven by Lévy noise when the drift and diffusion coefficients can grow super-linearly. As an application of our results, one can construct explicit Euler-type schemes for SDEs with delays (SDDEs) which are driven by Lévy noise and have super-linear coefficients. Strong convergence results are established and their rate of convergence is shown to be equal to that of the classical Euler scheme. It is proved that the optimal rate of convergence is achieved for $\mathcal{L}^2$-convergence which is consistent with the corresponding results available in the literature.
PRJul 7, 2017
Milstein-type Schemes of SDE Driven by Lévy Noise with Super-linear Diffusion CoefficientsChaman Kumar
We present a Milstein-type scheme for stochastic differential equations driven by Lévy noise with super-linear diffusion coefficients and establish its strong convergence.