L alpha-1 distance between two one-dimensional stochastic differential equations driven by a symmetric alpha-stable process

摘要

In this article, we consider a coefficient stability problem for one-dimensional stochastic differential equations driven by an alpha-stable process with alpha is an element of(1,2). More precisely, we find an upper bound for the L alpha-1(Omega,P) distance between two solutions in terms of the L alpha(R, mu(alpha)(x0)) distance of the coefficients for an appropriate measure mu(alpha)(x0) which characterizes symmetric stable laws and depends on the initial value of the stochastic differential equation. We obtain this result using the method introduced by Komatsu (Proc Jpn Acad Ser A Math Sci 58(8):353-356, 1982) which is used in the proof of uniqueness of solutions together with an upper bound for the transition density function of the solution of the stochastic differential equation obtained by Kulik (The parametrix method and the weak solution to an SDE driven by an alpha-stable noise. arXiv:1412.8732, 2014).