A Local Linearization (LL) method for the numerical integration of Random Differential Equations (RDE) is introduced. The classical LL approach is adapted to this type of equations, which are defined by random vector fields that are typically at most Hölder continuous with respect to the time argument. The order of strong convergence of the method is studied. It turns out that the LL method improves the order of convergence of conventional numerical methods that have been applied to RDEs. Additionally, the performance of the LL method is illustrated by means of numerical simulations, which show that it behaves well even in those equations with complicated noisy dynamics where conventional methods fail. (2000): 34F05, 34K28, 60H25. AMS subject classification

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.

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