Adams-type methods for the numerical solution of stochastic ordinary differential equations

The modelling of many real life phenomena for which either the parameter estimation is difficult, or which are subject to random noisy perturbations, is often carried out by using stochastic ordinary differential equations (SODEs). For this reason, in recent years much attention has been devoted to deriving numerical methods for approximating their solution. In particular, in this paper we consider the use of linear multistep formulae (LMF). Strong order convergence conditions up to order 1 are stated, for both commutative and non-commutative problems. The case of additive noise is further investigated, in order to obtain order improvements. The implementation of the methods is also considered, leading to a predictor-corrector approach. Some numerical tests on problems taken from the literature are also included.