We consider the filtering problem of estimating a hidden random variable X by noisy observations. The noisy observation process is constructed by a randomized Markov bridge (RMB) $(Z_t{)}_{t\in [0,T]}$ of which terminal value is set to $Z_T=X$. That is, at the terminal time T, the noise of the bridge process vanishes and the hidden random variable X is revealed. We derive the explicit filtering formula, governing the dynamics of the conditional probability process, for a general RMB. It turns out that the conditional probability is given by a function of current time t, the current observation $Z_t$, the initial observation $Z_0$, and the a priori distribution ν of X at t = 0. As an example for an RMB, we explicitly construct the skew-normal randomized diffusion bridge and show how it can be utilized to extend well-known commodity pricing models and how one may propose novel stochastic price models for financial instruments linked to greenhouse gas emissions.

我们考虑通过噪声观察估计隐藏的随机变量X的滤波问题。噪声观测过程由随机马尔可夫桥（RMB）构建$（{ž}_{Ť}{）}_{Ť\in [0，Ť]}$ 终端值设置为 ${ž}_{Ť}=X$。即，在终端时间T处，桥接过程的噪声消失，并且隐藏的随机变量X被揭示。我们推导了用于一般人民币的显式过滤公式，用于控制条件概率过程的动力学。事实证明，条件概率是由当前时间t，当前观测值的函数给出的${ž}_{Ť}$，初步观察 ${ž}_0$，以及X在t  = 0时的先验分布ν。以人民币为例，我们明确构建了偏正态随机扩散桥，并说明了如何利用它来扩展著名的商品定价模型以及为与温室气体排放有关的金融工具提出新颖的随机价格模型。