Counting processes (CPs) are important mathematical models with a variety of applications in signal processing, telecommunication, queuing, neuroscience, systems and synthetic biology, meteorology, insurance and finance. A CP$(Y_ {t})_{ t\geq 0}$can be characterized by its conditional intensity (CI), i.e., the$\sigma (Y_{s}, 0 \le s \le t)$-intensity, for which the filtration is generated by$(Y_ {t})_{ t\geq 0}$. The CI is the central quantity from which it is possible to compute information-theoretic measures, and the asymptotic of the CI suffices for their asymptotic versions. Two examples for such measures are the mutual information rate (MIR) of a signal and its Poisson channel output, and the relative entropy rate (RER) between CPs. In order to analytically access the asymptotic of the CI, we introduce the class of CP models for which the CI progresses as a piecewise-deterministic Markov process on an augmented state space and with deterministic jump sizes. This class includes Markov-modulated Poisson processes and self-exciting counting processes. For this class we derive the probability evolution equation of the CI and the fixed point equation for its embedded Markov chain. For the asymptotic CI distribution (ACID) we obtain an analytic description by stationary analysis methods for Markov processes. We present a simulation-free method to compute the ACID, when the dimension of the auxiliary state space is low. Using the ACID, we contribute a new method for the computation of the MIR of the Poisson channel as well as the RER between CPs; and suggest its use also for the empirical assessment of the similarity of CPs. We apply the technique to various naturally occurring CPs, such as the random telegraph modulated Poisson process and the Hawkes process.

A minimal model for gene expression, consisting of a switchable promoter together with the resulting messenger RNA, is equivalent to a Poisson channel with a binary Markovian input process. Determining its capacity is an optimization problem with respect to two parameters: the average sojourn times of the promoter's active (ON) and inactive (OFF) state. An expression for the mutual information is found by exploiting the link with filtering theory. For fixed peak power, three bandwidth-like constraints are imposed by lower-bounding (i) the average sojourn times (ii) the autocorrelation time and (iii) the average time until a transition. OFF-favoring optima are found for all three constraints, as commonly encountered for the Poisson channel. In addition, constraint (i) exhibits a region that favors the ON state, and (iii) shows ON-favoring local optima.