Weak Error on the densities for the Euler scheme of stable additive SDEs with Besov drift

TL;DR

The paper analyzes Euler discretization for multi-dimensional SDEs driven by symmetric α-stable noise with a distributional Besov drift. By leveraging a Besov-space framework, heat-kernel estimates, and Duhamel representations, the authors derive a weak-density convergence rate of (γ−ε)/α, where γ>0 encodes the drift-regularity and Krylov-type conditions. A careful error decomposition into six Δ-terms, combined with controlled Hölder-modulus bounds and a Grönwall-type argument, yields the rate and clarifies how parameter choices (α,β,p,q,r) influence convergence. The results extend weak-error-rate insights from Brownian and Lévy-driven settings to the challenging Besov-drift, multi-dimensional, strictly stable case, with potential extensions toBrownian noise under analogous heat-kernel estimates. The framework provides robust density-weak convergence results for discretizations under rough drifts, relevant for numerical approximation of stable-driven SDEs in high-dimensional settings.

Abstract

We are interested in the Euler-Maruyama dicretization of the formal SDE, $dX_t=b(t,X_t)dt+dZ_t$, where $Z$ is a symmetric isotropic d dimensional stable process of index $α\in (1,2)$, and $b$ is distributional. It belongs to a mix Lebesgue-Besov space. The associated parameters satisfy some constraints which guarantee weak-well posedness. Defining an appropriate Euler scheme, we obtain a convergence rate for the weak error on the densities. The rate depends on the parameters.

Figures (1)

• Figure 1: Decompositon of the error into 6 terms, including the first and last time steps ($\Delta_1$ and $\Delta_6$), the approximation error for the distribution $b$ ($\Delta_3$), the errors due to the (ir)regularity of the density and the noise ($\Delta_5$ and $\Delta_2$) and the term that is treated via a Gronwall-type argument ($\Delta_4$).

Theorems & Definitions (15)

• Lemma 1: Convolution properties and spatial moment for $\bar{p}_\alpha$
• Lemma 2: Stable sensitivities - Estimates on the $\alpha$-stable kernel ${p}_\alpha$
• Proposition 1: Heat kernel estimates for the densities
• Theorem 1: Convergence Rate for the stable-driven Euler scheme with Besov drift
• Remark 1: About test convergence rate
• Remark 2: About test functions.
• Lemma 3: Useful bounds for $\mathfrak{b}_h$
• proof
• Lemma 4: The approximate singular drift in the density of the driving noise
• proof