On the convergence of adaptive approximations for stochastic differential equations

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 instead use rough path theory to establish pathwise 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 adaptive step sizes to have a “no skip” property and to take values at only dyadic times. Secondly, in contrast to the Euler-Maruyama method, we require the SDE solver to have unbiased “Lévy 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 “Lévy 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.

💡 Research Summary

This paper addresses the long‑standing problem of proving convergence for stochastic differential equation (SDE) solvers that employ adaptive step sizes depending on future Brownian increments, i.e., non‑previsible step controls. Traditional mean‑square error analyses break down because the step size is no longer measurable with respect to the σ‑algebra generated by past information. Building on the pioneering work of Gaines and Lyons, the authors combine the Brownian‑tree data structure with rough‑path theory to obtain pathwise convergence results for a broad class of adaptive methods applied to Stratonovich SDEs with sufficiently smooth coefficients.

The key technical conditions are twofold. First, the adaptive partition must be “no‑skip”: each refinement halves a previous step and never jumps over already visited time points. Second, the partition must be dyadic (step sizes of the form 2⁻ⁿ) and its mesh must tend to zero almost surely. Under these assumptions, any numerical scheme whose one‑step update can be written as
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