We consider in this paper, a general class of stochastic differential equations driven by stable processes with Lipschitz drift  coefficients and non-Lipschitz diffusion coefficients. A strong Euler-Maruyama approximate  solution  is proved whenever the diffusion coefficient is  Hölder continuous with exponent satisfying some condition.  We derive also the  strong rate of convergence.

Our proposed method is new in this context and  based on  a truncation method by separating the big and small jumps  of the stable process in the Lévy-Itô decomposition. Along the paper we give some numerical simulation of stochastic models that match our results namely some  stable driven  Ornstein-Uhlenbeck and  Cox–Ingersoll–Ross processes.