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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.
A conjecture appears in cite{milsteinscheme}, in the form of a remark, where it is stated that it is possible to construct, in a specified way, any high order explicit numerical schemes to approximate the solutions of SDEs with superlinear coefficien
Motivated by the results of cite{sabanis2015}, we propose explicit Euler-type schemes for SDEs with random coefficients driven by Levy noise when the drift and diffusion coefficients can grow super-linearly. As an application of our results, one can
Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates f
The aim of this paper is to obtain convergence in mean in the uniform topology of piecewise linear approximations of Stochastic Differential Equations (SDEs) with $C^1$ drift and $C^2$ diffusion coefficients with uniformly bounded derivatives. Conver
We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in `{C}esaro sense. In such a case, the averaged coefficients could be disc