What is Conditional intensity, compensator?

The conditional intensity of a counting process is the adapted process $\lambda_t$ with $\P(N_{t+dt} - N_t = 1 \mid \mathcal F_t) = \lambda_t\,dt + o(dt)$: the instantaneous rate of an event given the past. Its compensator is $\Lambda_t = \int_0^t\lambda_s\,ds$, the predictable increasing process with $N_t - \Lambda_t$ a local martingale.