  For $\alpha \in (1,2)$, we study the following stochastic differential
equation driven by a non-degenerate symmetric $\alpha$-stable process in
${\mathbb R}^d$:
  \begin{align*} {\mathord{{\rm d}}} X_t=b(t,X_t){\mathord{{\rm d}}}
t+\sigma(t,X_{t-}){\mathord{{\rm d}}} L_t^{(\alpha)},\ \ X_0 =x \in {\mathbb
R}^d, \end{align*} where $b$ belongs to $ L^\infty({\mathbb R}_+;{\mathbf
B}_{\infty,\infty}^{-\beta}({\mathbb R}^d))$ with some
$\beta\in(0,\frac{\alpha-1}{2})$, and $\sigma:{\mathbb R}_+\times {\mathbb R}^d
\to {\mathbb R}^d \otimes {\mathbb R}^d$ is a $d \times d $ matrix-valued
measurable function. We point out that the noise could be a cylindrical
$\alpha$-stable process. We first show the generalized martingale problems and
then establish the stability estimates of solutions. As an application, we give
the weak convergence rate of the Euler scheme for additive noises with drift
coefficient $b=b(x)$.
