Weak error on the densities for the Euler scheme of stable additive SDEs with H{ö}lder drift

TL;DR

This work analyzes the weak error on densities for the Euler discretization of an SDE with additive $\alpha$-stable noise and a Hölder continuous drift. By employing a time-randomized Euler scheme and Duhamel representations, the authors exploit parabolic bootstrap effects to derive a density-based convergence rate of $h^{\gamma/\alpha}$, where $\gamma = \alpha + \beta - 1$. The analysis hinges on sharp density bounds for the stable noise, forward-time regularity, and a careful decomposition of the discretization error, culminating in a discrete Grönwall argument that yields the stated rate with a mild time-singularity factor $(1+t^{-\beta/\alpha})$. The results extend known Brownian-framework rates to stable-driven dynamics and provide a robust framework for weak-density convergence with rough drift, with implications for related weak-error estimates for broad test-function classes.

Abstract

We are interested in the Euler-Maruyama dicretization of the SDE dXt =b(t,Xt)dt+ dZt, X0 =x$\in$Rd, where Zt is a symmetric isotropic d-dimensional $α$-stable process, $α$ $\in$ (1, 2] and the drift b $\in$ L$\infty$ ([0,T],C$β$(Rd,Rd)), $β$ $\in$ (0,1), is bounded and H{ö}lder regular in space. Using an Euler scheme with a randomization of the time variable, we show that, denoting $γ$\,:= $α$ + $β$ -- 1, the weak error on densities related to this discretization converges at the rate $γ$/$α$.

Theorems & Definitions (14)

• Proposition 1: Density estimates for the diffusion and its Euler scheme
• Remark 1: About additional controls on the density of the SDE and the Euler scheme
• Theorem 1: Convergence Rate for the stable-driven Euler scheme with $L_t^\infty \mathcal{C}_x^{\beta}$ drift
• Remark 2: Weak error involving an additional test function
• Proposition 2: Density estimates for the heat equation
• proof
• Proposition 3: Duhamel representations for the densities of the SDE and the Euler scheme
• Lemma 1: Smoothing effect of the drift
• proof
• Lemma 2: Stable sensitivities - Estimates on the $\alpha$-stable kernel