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 proposed estimators.

翻译：卡尔曼和布西的工作建立了时间连续线性系统中滤波与最优估计之间的对偶性。这种对偶性最近已被扩展到时间连续非线性系统，其形式为受后向随机偏微分方程约束的优化问题。本文从适当的前向-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。同时证明，估计问题的某些特定公式化表述会导出类似于卡尔曼和布西最初研究的线性高斯情况下的确定性公式。最后，将部分观测扩散过程的最优控制作为所提估计器的应用进行讨论。