Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process

Abstract

We prove quantitative pathwise stability estimates for one-dimensional stochastic differential equations driven by a common Brownian motion and a common symmetric $α$-stable process, where $α\in(1,2)$. The comparison is made under a synchronous coupling and is measured by $\sup_{0\le t\le T}\mathbb E|X_t-\wt X_t|^{α-1}$. The perturbed drift and Brownian diffusion coefficients are spatially Lipschitz, while the perturbed stable jump coefficient may be Hölder continuous down to the critical exponent $1/α$. The estimate is expressed in terms of the initial error and three law-weighted coefficient errors: the drift error $B$, the Brownian diffusion error $A$, and the stable jump-coefficient error $S$. The Brownian component produces a second-order correction term in Itô's formula. This term is controlled by a second-derivative estimate for a Komatsu-type mollification of $|x|^{α-1}$. For stable jump coefficients with Hölder exponent $\wtη>1/α$, the estimate gives explicit power rates in $B,A,S$; at $\wtη=1/α$, it gives a logarithmic rate. The same method yields estimates with predictable forcing errors and time-uniform tail bounds via stopped quasi-martingales.