This thesis presents a comprehensive formulation of adaptive time-step numerical integrators for stochastic differential equations with additive noise, introducing new techniques to embed exponential-based schemes. It also introduces two embedded A-stable explicit numerical schemes based on the Local Linearization (LL) approach: the embedded LL and embedded LL Runge-Kutta schemes, which are entirely new in the literature. Additionally, we present numerical linear algebra techniques to optimize the Padé algorithm for computing matrix exponentials in our LL schemes, resulting in A-stable adaptive schemes with comparable (or even lower) computational costs than unstable adaptive schemes.