On the limit distribution for stochastic differential equations driven by cylindrical non-symmetric $\alpha$-stable L\'{e}vy processes

TL;DR

This paper studies the convergence of solutions to stochastic differential equations driven by cylindrical non-symmetric -stable processes to those driven by Brownian motion as  approaches 2, including convergence rates and practical examples.

Contribution

It establishes weak convergence and optimal rates for solutions of SDEs driven by -stable processes to Brownian-driven SDEs, extending understanding of limit distributions.

Findings

Weak convergence of solutions as  2

Derived explicit convergence rates depending on 2-

Applied results to a one-dimensional SDE, confirming optimality

Abstract

This article deals with the limit distribution for a stochastic differential equation driven by a non-symmetric cylindrical $\alpha$-stable process. Under suitable conditions, it is proved that the solution of this equation converges weakly to that of a stochastic differential equation driven by a Brownian motion in the Skorohod space as $\alpha\rightarrow2$. Also, the rate of weak convergence, which depends on $ 2-\alpha$, for the solution towards the solution of the limit equation is obtained. For illustration, the results are applied to a simple one-dimensional stochastic differential equation, which implies the rate of weak convergence is optimal.