On the limit distribution for stochastic differential equations driven by cylindrical non-symmetric $\alpha$-stable Lévy processes

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 $\alpha \to 2$. 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.