= Compensator of a counting process
{title2=$\Lambda(t)$}
= Counting-process compensator
{synonym}
A <Compensator of a counting process> $N$ is a <predictable process>[predictable] increasing process $\Lambda$, normalized by $\Lambda_0=0$, such that $N-\Lambda$ is a <local martingale>. Under the standard local-integrability and <usual conditions for a filtration>[usual-filtration conditions], the <predictable process>[predictable] <compensator of a counting process>[compensator] is unique up to <indistinguishability of stochastic processes>[indistinguishability]. It describes accumulated <conditional probability>[conditional] <event> rate, not an <event> <probability>.
For a rate-$\lambda$ <Poisson process>, $\Lambda_t=\lambda t$ and $N_t-\lambda t$ is a <martingale>. If $T$ is <exponential distribution>[exponential] of rate $\lambda$ and $N_t=\mathbf1_{\{T\leq t\}}$ counts a single <event>, its <predictable process>[predictable] <counting-process intensity>[intensity] is $\lambda\mathbf1_{\{N_{t-}=0\}}$ and
$$
\Lambda_t=\lambda(t\wedge T).
$$
The rate shuts off after the <event>. This pathwise accumulated <counting-process intensity>[intensity] can exceed one even before an <event> occurs; it differs from the deterministic <probability> $\mathbb P(T\leq t)=1-e^{-\lambda t}$.
A <compensator of a counting process>[compensator] need not be <absolute continuity of measures>[absolutely continuous] in time. The deterministic count $N_t=\mathbf1_{\{t\geq1\}}$ has <compensator of a counting process>[compensator] $\Lambda=N$, because its jump time is known in advance and $N-\Lambda=0$. No ordinary <Rate function> integrated against $dt$ produces that step. When $\Lambda_t=\int_0^t\lambda_sds$, the <predictable process>[predictable] <derivative> $\lambda_s$ is the <counting-process intensity>. For an observed survival <event> under <independent censoring> it takes the familiar form $\lambda_s=Y_s h_s$.
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