In this paper, we consider the development of unbiased estimators for the ensemble Kalman−Bucy filter (EnKBF). The EnKBF is a continuous-time filtering methodology, which can be viewed as a continuous-time analog of the famous discrete-time ensemble Kalman filter. Our unbiased estimators will be motivated from recent work (Rhee and Glynn, Oper. Res., 63:1026−1053, 2015) which introduces randomization as a means to produce unbiased and finite variance estimators. The randomization enters through both the level of discretization and through the number of samples at each level. Our unbiased estimator will be specific to models that are linear and Gaussian. This is due to the fact that the EnKBF itself is consistent, in the large particle limit N → ∞, with the Kalman−Bucy filter, which allows us one derive theoretical insights. Specifically, we introduce two unbiased EnKBF estimators that will be applied to two particular variants of the EnKBF, which are the deterministic and vanilla EnKBF. Numerical experiments are conducted on a linear Ornstein−Uhlenbeck process, which includes a high-dimensional example. Our unbiased estimators will be compared to the multilevel. We also provide a proof of the multilevel deterministic EnKBF, which provides a guideline for some of the unbiased methods.

中文翻译：

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普通和确定性集合卡尔曼-布西滤波器的无偏估计

在本文中，我们考虑开发集成卡尔曼-布西滤波器（EnKBF）的无偏估计器。EnKBF 是一种连续时间滤波方法，可以看作是著名的离散时间集成卡尔曼滤波器的连续时间模拟。我们的无偏估计量将受到最近工作的启发（Rhee and Glynn, Oper. Res., 63:1026−1053, 2015），该工作引入随机化作为产生无偏和有限方差估计量的方法。随机化通过离散化级别和每个级别的样本数量进行。我们的无偏估计器将特定于线性和高斯模型。这是因为 EnKBF 本身在大粒子极限 N → ∞ 下与 Kalman−Bucy 滤波器是一致的，这使我们能够得出理论见解。具体来说，我们引入了两个无偏 EnKBF 估计器，它们将应用于 EnKBF 的两个特定变体，即确定性 EnKBF 和普通 EnKBF。数值实验是在线性 Ornstein−Uhlenbeck 过程上进行的，其中包括一个高维示例。我们的无偏估计量将与多层次进行比较。我们还提供了多级确定性 EnKBF 的证明，它为一些无偏方法提供了指导。