In this paper, we consider nonlinear stochastic systems and intersect ideas from nonlinear control theory and numerical analysis. In particular, we use the idea of relative degree. This concept guarantees smoothness properties of the output and this, in turn, allows one to establish properties that are unique to the control-theoretic perspective. The contributions of the current paper are threefold. Firstly, we define different error measures that extend the ideas of local and global approximation errors for nonlinear stochastic systems. Secondly, we demonstrate that the concept of relative degree plays a key role in obtaining higher order of accuracy for integration procedures compared to Euler-Maruyama integration. We show that a particular state-space model, named STTS model, has an improved order of accuracy when compared to an Euler-Maruyama approximation, at no significant extra computational cost. Thirdly, we show that a further approximation to the STTS model, named MSTTS model, while retaining the order of local errors, has explicit sampling zero dynamics, associated with the noise processes, that have no continuous-time counterpart. The extra zero dynamics are shown to be a function of the Euler-Frobenius polynomials. To the best of the authors' knowledge, this is the first reference to sampling zero dynamics for stochastic nonlinear systems.