Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

TL;DR

This paper presents an explicit adaptive Milstein method for SDEs with monotone coefficients, achieving strong convergence of order one and effectively managing step sizes near boundary regions.

Contribution

It introduces a novel adaptive Milstein scheme for SDEs with monotone coefficients, ensuring strong convergence without commutativity assumptions.

Findings

Achieves strong $L_2$ convergence of order one.

Backstop method probability can be minimized.

Outperforms fixed-step Milstein variants on tests.

Abstract

We introduce an explicit adaptive Milstein method for stochastic differential equations (SDEs) with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly $L_2$ convergent of order one. This order is inherited by an explicit adaptive Euler-Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.