Compensator - crowlogic/arb4j GitHub Wiki

The compensator of a counting process is defined as the integral of the intensity function $\lambda(t)$, and it is denoted by $\Lambda(t)$. The intensity function $\lambda(t)$ gives the instantaneous rate at which events occur at time $t$, and the compensator $\Lambda(t)$ is chosen so that the counting process $N(t) - \Lambda(t)$ is a martingale, where $N(t)$ represents the number of events that have occurred up to time $t$.

More specifically, the compensator $\Lambda(t)$ is defined as:

$$ \Lambda(t) = \int_0^t \lambda(s) ds $$

where $\lambda(s)$ is the intensity function at time $s$. Then, the martingale property of $N(t) - \Lambda(t)$ can be expressed as:

$$ E[N(t) - \Lambda(t) \mid \mathcal{F}(s)] = N(s) - \Lambda(s) $$

for all $s < t$, where $\mathcal{F}(s)$ denotes the information available up to time $s$.

The compensator $\Lambda(t)$ is a useful tool in the analysis of counting processes, as it allows us to model the underlying intensity function in a way that ensures certain desirable properties, such as stationarity and independence of increments. It also allows us to estimate the intensity function from observed event times, using methods such as maximum likelihood estimation or nonparametric methods based on kernel smoothing.