Phase transition in the EM scheme of an SDE driven by $\alpha$stable noises with $\alpha \in (0,2]$
Abstract
We study in this paper the EM scheme for a family of wellposed critical SDEs with the drift $x\log(1+x)$ and $\alpha$stable noises. Specifically, we find that when the SDE is driven by a rotationally symmetric $\alpha$stable processes with $\alpha=2$ (i.e. Brownian motion), the EM scheme is bounded in the $L^2$ sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric $\alpha$stable process with $\alpha \in (0,2)$, all the $\beta$th moments, with $\beta \in (0,\alpha)$, of the EM scheme blow up. This demonstrates a phase transition phenomenon as $\alpha \uparrow 2$. We verify our results by simulations.
 Publication:

arXiv eprints
 Pub Date:
 March 2024
 DOI:
 10.48550/arXiv.2403.18626
 arXiv:
 arXiv:2403.18626
 Bibcode:
 2024arXiv240318626W
 Keywords:

 Mathematics  Probability