Compensator of a counting process ($\Lambda (t)$)

A Compensator of a counting process $N$ is a predictable increasing process $\Lambda$, normalized by ${\Lambda }_0=0$, such that $N-\Lambda$ is a local martingale. Under the standard local-integrability and usual-filtration conditions, the predictable compensator is unique up to indistinguishability. It describes accumulated 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 of rate $\lambda$ and $N_t=1_{\{T\le t\}}$ counts a single event, its predictable intensity is $\lambda 1_{\{N_{t-}=0\}}$ andThe rate shuts off after the event. This pathwise accumulated intensity can exceed one even before an event occurs; it differs from the deterministic probability $\mathbb{P}(T\le t)=1-e^{-\lambda t}$.

A compensator need not be absolutely continuous in time. The deterministic count $N_t=1_{\{t\ge 1\}}$ has 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 ${\Lambda }_t={\int }_0^t{\lambda }_{s}ds$, the predictable derivative ${\lambda }_s$ is the counting-process intensity. For an observed survival event under independent censoring it takes the familiar form ${\lambda }_s=Y_s{h}_s$.

If a counting process $N$ has predictable conditional intensity $\lambda$, subtracting its compensator of a counting process gives the associated martingale $M$. Conditional expected event increments and compensator increments agree, so their difference is conditionally centered. With the necessary integrability, $M$ is a mean-zero martingale; for a simple no-ties process its predictable quadratic variation is ${\int }_0^t\lambda (u)du$. A predictable weighted martingale integral has mean zero and variance determined by the square of its weights. In survival analysis, $\lambda =Yh$ produces the event-minus-expected representation used to derive the Nelson–Aalen estimator.