The connection between forward backward doubly stochastic differential equations and the optimal filtering problem is established without using the Zakai's equation. The solutions of forward backward doubly stochastic differential equations are expressed in terms of conditional law of a partially observed Markov diffusion process. It then follows that the adjoint time-inverse forward backward doubly stochastic differential equations governs the evolution of the unnormalized filtering density in the optimal filtering problem.

中文翻译：

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前向后向双随机微分方程与扩散过程的最优滤波

前向后向双随机微分方程与最优滤波问题的联系是在不使用Zakai方程的情况下建立的。前向后向双随机微分方程的解用部分观测马尔可夫扩散过程的条件定律表示。然后，伴随的时间逆前向后向双随机微分方程控制了最优滤波问题中非归一化滤波密度的演化。