Continuous Extension of Euler-Maruyama Method for Stochastic Differential Equations

In this work we focus on the development of continuous extension of Euler-Maruyama method, which is used to numerically approximate the solution of Stochastic Differential Equations (SDEs). We aim to provide an approximation of a given SDE in terms of a piecewise polynomial, because, as it is known in the deterministic case, a dense output allows to provide a more efficient error estimate and it is very effective for a variable step-size implementation. Hence, this contribution aims to provide a first building block in such directions, consisting in the development of the scheme.