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For $\alpha \in (1,2)$, we study the following stochastic differentialequation 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 {\mathbbR}^d, \end{align*} where $b$ belongs to $ L^\infty({\mathbb R}_+;{\mathbfB}_{\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-valuedmeasurable function. We point out that the noise could be a cylindrical$\alpha$-stable process. We first show the generalized martingale problems andthen establish the stability estimates of solutions. As an application, we givethe weak convergence rate of the Euler scheme for additive noises with driftcoefficient $b=b(x)$.

SDE driven by cylindrical $\alpha$-stable process with distributional drift and application