Despite the success of adaptive time-stepping in ODE simulation, it has so far seen few applications for Stochastic Differential Equations (SDEs). To simulate SDEs adaptively, methods such as the Virtual Brownian Tree (VBT) have been developed, which can generate Brownian motion (BM) non-chronologically. However, in most applications, knowing only the values of Brownian motion is not enough to achieve a high order of convergence; for that, we must compute time-integrals of BM such as $\int_s^t W_r \, dr$. With the aim of using high order SDE solvers adaptively, we extend the VBT to generate these integrals of BM in addition to the Brownian increments. A JAX-based implementation of our construction is included in the popular Diffrax library (https://github.com/patrick-kidger/diffrax). Since the entire Brownian path produced by VBT is uniquely determined by a single PRNG seed, previously generated samples need not be stored, which results in a constant memory footprint and enables experiment repeatability and strong error estimation. Based on binary search, the VBT's time complexity is logarithmic in the tolerance parameter $\varepsilon$. Unlike the original VBT algorithm, which was only precise at some dyadic times, we prove that our construction exactly matches the joint distribution of the Brownian motion and its time integrals at any query times, provided they are at least $\varepsilon$ apart. We present two applications of adaptive high order solvers enabled by our new VBT. Using adaptive solvers to simulate a high-volatility CIR model, we achieve more than twice the convergence order of constant stepping. We apply an adaptive third order underdamped or kinetic Langevin solver to an MCMC problem, where our approach outperforms the No U-Turn Sampler, while using only a tenth of its function evaluations.

翻译：尽管自适应时间步长在常微分方程模拟中取得了成功，但目前在随机微分方程中的应用仍十分有限。为实现对SDE的自适应模拟，研究者已开发出虚拟布朗树等方法，可非时序地生成布朗运动。然而多数应用中，仅获取布朗运动数值不足以达到高收敛阶——为此必须计算如$\int_s^t W_r \, dr$等布朗运动时间积分。为自适应使用高阶SDE求解器，我们在VBT基础上扩展了这些积分的生成能力。基于JAX的实现已集成至流行的Diffrax库（https://github.com/patrick-kidger/diffrax）。由于VBT生成的完整布朗路径由单一PRNG种子唯一确定，无需存储先前生成的样本，从而保持恒定内存占用，并实现实验可重复性与强误差估计。基于二分查找的VBT时间复杂度呈对数级$\varepsilon$。与原始VBT算法仅保证二进制定时精度的局限不同，我们证明：在任意查询时间间隔不小于$\varepsilon$时，新构造精确匹配布朗运动及其时间积分的联合分布。本文展示了两种由新型VBT赋能的自适应高阶求解器应用：通过自适应求解器模拟高波动CIR模型，收敛阶较恒定步长提升两倍以上；将自适应三阶阻尼/动能朗之万求解器应用于MCMC问题，仅用No U-Turn采样器十分之一函数评估次数即实现性能超越。