Continuous time random walks and queues: Explicit forms and approximations of the conditional law with respect to local times

In the filtering problem considered here, the state process is a continuous time random walk and the observation process is an increasing process depending deterministically on the trajectory of the state process. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to a rescaled M/M/1 queueing model. In both these cases, the sequence of the observation processes converges to a local time, and a convergence result for the respective filters is given. The case of a queueing model when the observation is the idle time is also considered.