  In this paper, we study numerical approximations for stochastic differential
equations (SDEs) that use adaptive step sizes. In particular, we consider a
general setting where decisions to reduce step sizes are allowed to depend on
the future trajectory of the underlying Brownian motion. Since these adaptive
step sizes may not be previsible, the standard mean squared error analysis
cannot be directly applied to show that the numerical method converges to the
solution of the SDE. Building upon the pioneering work of Gaines and Lyons, we
shall instead use rough path theory to establish convergence for a wide class
of adaptive numerical methods on general Stratonovich SDEs (with sufficiently
smooth vector fields). To our knowledge, this is the first convergence
guarantee that applies to standard solvers, such as the Milstein and Heun
methods, with non-previsible step sizes. In our analysis, we require the
sequence of adaptive step sizes to be nested and the SDE solver to have
unbiased "L\'evy area" terms in its Taylor expansion. We conjecture that for
adaptive SDE solvers more generally, convergence is still possible provided the
method does not introduce "L\'evy area bias". We present a simple example where
the step size control can skip over previously considered times, resulting in
the numerical method converging to an incorrect limit (i.e. not the
Stratonovich SDE). Finally, we conclude with an experiment demonstrating the
accuracy of Heun's method and a newly introduced Splitting Path-based
Runge-Kutta scheme (SPaRK) when used with adaptive step sizes.
