Counting processes

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.

Discrete distributions derived from renewal processes, i.e. distributions of the number of events by some time t are beginning to be used in econometrics and health sciences. A new fast method is presented for computation of the probabilities for these distributions. We calculate the count probabilities by repeatedly convolving the discretized distribution, and then correct them using Richardson extrapolation. When just one probability is required, a second algorithm is described, an adaptation of De Pril's method, in which the computation time does not depend on the ordinality, so that even high-order probabilities can be rapidly found. Any survival distribution can be used to model the inter-arrival times, which gives a rich class of models with great flexibility for modeling both underdispersed and overdispersed data. This work could pave the way for the routine use of these distributions as an additional tool for modeling event count data. An empirical example using fertility data illustrates the use of the method and was fully implemented using an R (R Core Team, 2015) package Countr (Baker et al., 2016) developed by the authors and available upon request.