$L^{\alpha-1}$ distance between two one-dimensional stochastic differential equations with drift terms driven by a symmetric $\alpha$-stable process

TL;DR

This paper develops a stability theory for one-dimensional SDEs driven by symmetric alpha-stable processes, providing explicit convergence rates and a novel coefficient distance measurement method.

Contribution

It introduces a new weighted integral norm for coefficients and establishes the first explicit convergence rates for stability of such SDEs with time-dependent coefficients.

Findings

Established a Hölder-type estimate for the $L^{eta}( ext{Ω})$ distance between solutions.

Introduced a weighted integral norm based on transition densities for measuring coefficient differences.

Derived convergence rates in probability for solution paths.

Abstract

This paper establishes a quantitative stability theory for one-dimensional stochastic differential equations (SDEs) with non-zero drift, driven by a symmetric $\alpha$-stable process for $\alpha\in(1,2)$. Our work generalizes the classical pathwise comparison method, pioneered by Komatsu for uniqueness problems, to address the stability of SDEs featuring both non-zero drift and, crucially, time-dependent coefficients. We provide the first explicit convergence rates for this broad class of SDEs. The main result is a H\"older-type estimate for the $L^{\alpha-1}(\Omega)$ distance between two solution paths, which quantifies the stability with respect to the initial values and coefficients. A key innovation of our approach is the measurement of the distance between coefficients. Instead of using a standard supremum norm, which would impose restrictive conditions, we introduce a weighted integral norm constructed from the transition probability density of the baseline solution. This technique, which generalizes the framework of Nakagawa \cite{Nakagawa}, is essential for handling time-dependent perturbations and effectively localizes the error analysis. The proof is based on a refined analysis of a mollified auxiliary function, for which we establish a new, sharper derivative estimate to control the drift terms. Finally, we apply these stability results to derive corresponding convergence rates in probability, providing an upper bound for the tail probability of the uniform distance between solution paths.