We introduce a predictor-corrector discretisation scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler-Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler-Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles and noisier systems.

翻译：我们针对一类随机微分方程的数值积分提出了一种预测-校正离散化格式，并证明其具有弱1.0阶收敛性。该格式的关键特征在于其能够沿解的状态空间维度进行序贯（递归）构建，因此适用于高维状态空间模型的近似。以随机洛伦兹96系统作为测试模型，我们表明所提方法相比标准欧拉-丸山格式可采用更大的时间步长，从而以更低计算代价生成有效近似。我们还引入了对该预测-校正格式在作为连续时间系统离散时间贝叶斯滤波器构建模块时产生的误差的理论分析。最后，我们评估了若干采用所提序贯预测-校正欧拉格式与标准欧拉-丸山方法的集合卡尔曼滤波器的性能。数值实验表明，采用新序贯格式的滤波器可在更大时间步长、更小蒙特卡罗集合规模及更高噪声系统条件下有效运行。