The work of Kalman and Bucy has established a duality between filtering and optimal estimation in the context of time-continuous linear systems. This duality has recently been extended to time-continuous nonlinear systems in terms of an optimization problem constrained by a backward stochastic partial differential equation. Here we revisit this problem from the perspective of appropriate forward-backward stochastic differential equations. This approach sheds new light on the estimation problem and provides a unifying perspective. It is also demonstrated that certain formulations of the estimation problem lead to deterministic formulations similar to the linear Gaussian case as originally investigated by Kalman and Bucy. Finally, optimal control of partially observed diffusion processes is discussed as an application of the newly proposed estimators.

翻译：卡尔曼和布西的研究在时间连续线性系统中建立了滤波与最优估计之间的对偶性。近期，这一对偶性通过一个受后向随机偏微分方程约束的优化问题被推广到时间连续非线性系统。本文从恰当的前向-后向随机微分方程视角重新审视该问题。该方法为估计问题提供了新的见解，并给出了统一的视角。同时，本文还证明，某些估计问题的公式化表达会导出与卡尔曼和布西最初研究的线性高斯情形相似的确定性形式。最后，作为新提出估计器的应用实例，本文讨论部分观测扩散过程的最优控制问题。