Strong convergence rate of Euler-Maruyama method for stochastic differential equations with Hölder continuous drift coefficient driven by symmetric $α$-stable process
Wei Liu
TL;DR
This work addresses numerical approximation of SDEs driven by symmetric $α$-stable noise with Hölder continuous drift. It develops an alternative direct analysis based on Hölder and Bihari inequalities to establish the strong $L^p$ convergence of the Euler–Maruyama method without relying on the Kolmogorov equation, under the conditions $α∈(1,2)$ and $β∈(0, α/2)$ with $2β<α$. For $p∈(0,2]$, the error satisfies $\sup_{0≤t≤T} \mathbb{E}|x(t)-Y(t)|^p ≤ Δ^{pβ/α} C_5^{p/2} e^{C_6 Tp/2}$, with explicitly defined constants $C_5$ and $C_6$ depending on $K$, $K_2$, $T$, and $β$. A scalar numerical example demonstrates the method and illustrates how heavier tails (smaller $α$) yield larger jumps, validating the theoretical rate and highlighting the approach’s practical relevance. The results extend EM convergence theory to non-Lipschitz drift with additive α-stable noise and offer a framework potentially extendable to multi-dimensional or multiplicative settings.
Abstract
Euler-Maruyama method is studied to approximate stochastic differential equations driven by the symmetric $α$-stable additive noise with the $β$ Hölder continuous drift coefficient. When $α\in (1,2)$ and $β\in (0,α/2)$, for $p \in (0,2]$ the $L^p$ strong convergence rate is proved to be $pβ/α$. The proofs in this paper are extensively based on Hölder's and Bihari's inequalities, which is significantly different from those in Huang and Liao (2018).