By invoking a pathwise series expansion of Brownian motion, we propose to approximate a stochastic differential equation (SDE) with an ordinary differential equation (ODE). This allows us to reformulate Bayesian inference for a SDE as the parameter estimation task for an ODE. Unlike a nonlinear SDE, the likelihood for an ODE model is tractable and its gradient can be obtained using adjoint sensitivity analysis. This reformulation allows us to use an efficient sampler, such as NUTS, that rely on the gradient of the log posterior. Applying the reparameterisation trick, variational inference can also be used for the same estimation task. We illustrate the proposed method on a variety of SDE models. We obtain similar parameter estimates when compared to data augmentation techniques.