We develop a Bayesian inference method for discretely-observed stochastic differential equations (SDEs). Inference is challenging for most SDEs, due to the analytical intractability of the likelihood function. Nevertheless, forward simulation via numerical methods is straightforward, motivating the use of approximate Bayesian computation (ABC). We propose a conditional simulation scheme for SDEs that is based on lookahead strategies for sequential Monte Carlo (SMC) and particle smoothing using backward simulation. This leads to the simulation of trajectories that are consistent with the observed trajectory, thereby increasing the ABC acceptance rate. We additionally employ an invariant neural network, previously developed for Markov processes, to learn the summary statistics function required in ABC. The neural network is incrementally retrained by exploiting an ABC-SMC sampler, which provides new training data at each round. Since the SDE simulation scheme differs from standard forward simulation, we propose a suitable importance sampling correction, which has the added advantage of guiding the parameters towards regions of high posterior density, especially in the first ABC-SMC round. Our approach achieves accurate inference and is about three times faster than standard (forward-only) ABC-SMC. We illustrate our method in four simulation studies, including three examples from the Chan-Karaolyi-Longstaff-Sanders SDE family.

翻译：针对离散观测的随机微分方程（SDEs），我们提出了一种贝叶斯推断方法。由于似然函数的解析不可解性，大多数SDE的推断具有挑战性。然而，通过数值方法进行前向模拟是直接的，这促使我们采用近似贝叶斯计算（ABC）。我们提出了一种基于序贯蒙特卡洛（SMC）前瞻策略与后向模拟粒子平滑的SDE条件模拟方案。该方案通过生成与观测轨迹一致的轨迹路径，提高了ABC接受率。此外，我们采用先前为马尔可夫过程开发的不变神经网络，学习ABC所需的充分统计量函数。通过利用ABC-SMC采样器，该神经网络可在每轮训练中获取新数据并进行增量重训练。由于SDE模拟方案不同于标准前向模拟，我们提出了一种适当的重要性采样校正方法，该方法具有引导参数向高后验密度区域移动的额外优势，尤其在首个ABC-SMC轮次中效果显著。我们的方法实现了精确推断，且速度比标准（仅前向）ABC-SMC约快三倍。通过四项仿真研究（包括三个来自Chan-Karaolyi-Longstaff-Sanders SDE族的实例）验证了方法的有效性。