Convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise

There is a variety of strong Local Linearization (LL) schemes for the numerical integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is used in the numerical implementation of the strong Local Linear discretization. However, in contrast with the Local Linear discretization, the convergence rate of the LL schemes has not been studied so far. In this paper, two general theorems about this matter are presented and, with their support, additional results are derived for some particular schemes. As a direct application, the convergence rate of some strong LL schemes for SDEs with jumps is briefly expounded as well.