Filtering of a reflected Brownian motion with respect to its local time

We consider a filtering problem when the state process is a reflected Brownian motion Xt and the observation process is its local time Λs, for s ≤ t. For this model we derive an approximation scheme based on a suitable interpolation of the observation process Λt. The convergence of the approximating filter to the original one combined with an explicit con-struction of the approximating filter allows us to derive the explicit form of the original filter. The last result can be obtained also by means of the Azéma martingale