Stepsize control for the Milstein scheme using first-exit-times

We introduce a variable timestepping algorithm for the Milstein scheme applied to SDEs with a scalar stochastic forcing. Multiple local error estimates are used, corresponding to different terms appearing in the Taylor series of the local truncation error. The timesteps are then chosen so as to bound the standard deviation of the contribution of each of these terms over a unit time interval, analogous to the Error-PerUnit-Step control used for ODE solvers. The problem of determining the optimal timestep then translates to one of finding the first-exit distribution of the Wiener process from a simple boundary and eliminates the need for timestep rejections.