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.

翻译：我们引入了一种明确的适应性米尔斯坦法,用于不具有交流性条件的随机差异方程式(SDEs) 。漂移和扩散是单独在Lipschitz 的地方,并且一起满足一个单质条件。这种方法依赖于一系列路径定时战略,这些战略通过减少解决办法的步态接近一个球体的边界而发挥作用,如果时间步步太小,则采用支持性方法。我们证明,这种方案非常合一。在添加噪音的情况下,这种顺序是由一个明确的适应性欧拉-丸山方案继承的。此外,我们还表明,在任何阶段使用后站方法的可能性都可以被任意地缩小。我们将我们的方法与其他固定的米尔斯坦变体在一系列测试问题上进行比较。