An exponential split-step double balanced \(\vartheta \) Milstein scheme for SODEs with locally Lipschitz continuous coefficients

In current work, an exponential split-step double balanced \(\vartheta \) $\vartheta$ Milstein scheme has been suggested for SODEs with locally Lipschitz continuous coefficients. We have been investigated the moment boundedness and strong convergence of the scheme. Moreover, we prove that the convergence rate of the provided method is equal to 1.0. Under some conditions, exponential mean square stability and mean square contraction of our scheme have been studied. Numerical examples are provided to illustrate the theoretical results.