Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients

A new class of explicit Euler 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 probability and in ${\mathcal{L}}^p$ to the solution of the corresponding SDEs. Moreover, rate of convergence estimates are provided for ${\mathcal{L}}^p$ and almost sure convergence. In particular, the strong order $1\mathrm{/}2$ is recovered in the case of uniform ${\mathcal{L}}^p$-convergence.