Stochastic modelling with randomized Markov bridges

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) of which terminal value is set to . 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 , the initial observation , and the a priori distribution nu 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.