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

Yu Wang, Yimin Xiao, Lihu Xu

Abstract

We study in this paper the EM scheme for a family of well-posed critical SDEs with the drift $-x\log(1+|x|)$ 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 $L^2$ sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric $α$-stable process with $α\in (0,2)$, all the $β$-th moments, with $β\in (0,α)$, of the EM scheme blow up. This demonstrates a phase transition phenomenon as $α\uparrow 2$. We verify our results by simulations.

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

Abstract

We study in this paper the EM scheme for a family of well-posed critical SDEs with the drift and -stable noises. Specifically, we find that when the SDE is driven by a rotationally symmetric -stable processes with (i.e. Brownian motion), the EM scheme is bounded in the sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric -stable process with , all the -th moments, with , of the EM scheme blow up. This demonstrates a phase transition phenomenon as . We verify our results by simulations.
Paper Structure (8 sections, 7 theorems, 110 equations, 2 figures, 3 tables)

This paper contains 8 sections, 7 theorems, 110 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The drift $-x \log(1+|x|)$ and $\alpha \in (0,2]$: phase transition
  3. The polynomial growth drift and $\alpha \in (0,2)$: blow up
  4. Preliminary and Auxiliary Lemmas
  5. EM scheme in the case of $\alpha=2$
  6. EM scheme in the case of $\alpha \in (0,2)$
  7. Simulations
  8. Appendix: The existence and uniqueness of strong solution

Key Result

Theorem 1.1

Consider the EM scheme eq:critical EM with $\alpha=2$, i.e. the driven noise is Brownian motion. Then, for any fixed initial value $x_0$, there exist constants $\eta_0 \leqslant \min\{(1+\left|x_0\right|)^{-2}, \mathrm{e}^{-5}\}$ and $C>0$ such that for all $\eta \in (0, \eta_0]$,

Figures (2)

  • Figure 1: As $T = 10$, simulations values of the second absolute moment $\mathbb{E}\left|Y_k\right|^2$ for the EM scheme \ref{['EM scheme BM']} with initial $Y_0 = 1, 5, 10$, $\eta = 0.001$ and iteration steps $n = 10\,000$, $0\leqslant k \leqslant n$.
  • Figure 2: As $T = 100$, simulations values of the second absolute moment $\mathbb{E}\left|Y_k\right|^2$ for the EM scheme \ref{['EM scheme BM']} with initial $Y_0 = 1, 5, 10$, $\eta = 0. 01$ and iteration steps $n = 10\,000$, $0\leqslant k \leqslant n$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['Lemma 2.1']}
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['uniform bound of EM_BM']}
  • proof : Proof of Theorem \ref{['thm 2']}
  • ...and 3 more