CountingProcess

A counting process is a stochastic process that counts the number of events that occur over time defined as a sequence of non-negative integer-valued random variables $N_t$, where $N_t$ represents the number of events that have occurred up to time $t$.

Counting processes can model a wide range of phenomena because of their important ability to model events that occur randomly and at unpredictable times

A key component of a counting process is its intensity function $\lambda(t)$, which gives the instantaneous rate at which events occur at time $t$. In other words, $\lambda(t)$ represents the probability that an event occurs in the infinitesimal time interval $(t, t + dt]$, given that no event has occurred up to time $t$.

The intensity function plays a crucial role in the analysis of counting processes, as it determines the distribution of durations between successive events, and it can be used to compute various quantities of interest, such as the probability of observing a certain number of events in a given time interval or the expected time until the next event occurs .