Rates of convergence for multivariate SDEs driven by Lévy processes in the small-time stable domain of attraction

2025/09/15 by Cázares, Jorge González, Kramer-Bang, David
#60F05 #60G51. Secondary: 60F25 #FOS: Mathematics #Primary: 37H10 #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.12377

Abstract

This paper explores the rates of convergence of solutions for multivariate stochastic differential equations (SDEs) driven by Lévy processes within the small-time stable domain of attraction (DoA). Explicit bounds are derived for the uniform Wasserstein distance between solutions of two Lévy-driven SDEs, expressed in terms of driver characteristics. These bounds establish convergence rates in probability for drivers in the DoA, and yield uniform Wasserstein distance convergence for SDEs with additive noise. The methodology uses two couplings for Lévy driver jump components, leading to sharp convergence rates tied to the processes' intrinsic properties.

Citations

• Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs
• Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction
• Total variation distance between SDEs with stable noise and Brownian motion
• Subexponential lower bounds for f-ergodic Markov processes
• Optimal Wasserstein-1 distance between SDEs driven by Brownian motion and stable processes
• Optimal Markovian coupling for finite activity Lévy processes
• Invariant measures of Lévy-type operators and their associated Markov processes
• First order convergence of weak Wong--Zakai approximations of Lévy driven Marcus SDEs
• Zooming in on a Lévy process at its supremum
• On the maximal inequalities of Burkholder, Davis and Gundy
• Mean Field Forward-Backward Stochastic Differential Equations
• Meromorphic Levy processes and their fluctuation identities

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• Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises
• Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise