Phase transition in the EM scheme of an SDE driven by α-stable noises with α ∈ (0,2]

We study in this paper the EM scheme for a family of well-posed critical SDEs with the drift -xlog(1+|x|) and α-stable noises. Specifically, we find that when the SDE is driven by a rotationally symmetric α-stable processes with α=2 (i.e. Brownian motion), the EM scheme is bounded in the L² sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric α-stable process with α ∈ (0,2), all the β-th moments, with β ∈ (0,α), of the EM scheme blow up. This demonstrates a phase transition phenomenon as α ↑ 2. We verify our results by simulations.