Convergence of a partially truncated Euler-Maruyama method for SDEs with super-linear piecewise continuous drift and Hölder diffusion coefficients

The main purpose of this paper is to develop and analyze a partially truncated Euler-Maruyama method for numerically solving SDEs with super-linear piecewise continuous drift coefficients and \(\varvec{(1/2+\alpha )}\) $\mathbf{(}\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{+}\mathit{\alpha }\mathbf{)}$ -Hölder diffusion coefficients (PTEMH), for \(\varvec{\alpha \in [0,1/2]}\) $\mathit{\alpha }\mathbf{\in }\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{]}$ . We first present an analytical form for the unique solution of such problems. Then we establish the strong convergence theory of the PTEMH scheme. We show that the convergence rate of the proposed method in the case \(\varvec{\alpha \in (0,1/2]}\) $\mathit{\alpha }\mathbf{\in }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{]}$ reaches \(\varvec{\alpha }\) $\mathit{\alpha }$ , which is optimal compared to the explicit Euler-Maruyama method. Finally, numerical results are given to confirm the theoretical convergence rate.