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.

翻译：卡尔曼和布西的工作确立了时间连续线性系统中滤波与最优估计之间的对偶性。近年来，这种对偶性已通过受后向随机偏微分方程约束的优化问题，被推广至时间连续非线性系统。本文从合适的前向-后向随机微分方程视角重新审视该问题。这一方法为估计问题提供了新的启示，并形成了统一的理论视角。同时，研究表明，估计问题的某些公式化表述可导出与卡尔曼和布西最初研究的线性高斯情形类似的确定性表述。