Counting-process intensity in survival analysis ($\mathbb{E}(d{N}_i\mid {\mathcal{H}}_{t-})=Y_i{h}_{i}dt$)

An observed event counting process has conditional increment $\mathbb{E}(d{N}_i(t)\mid {\mathcal{H}}_{t-})=Y_i(t)h_i(t)dt$, where $Y_i$ is the at-risk process. The filtration records the past observed events, censoring, and covariates; the integral of $Y_i{h}_i$ is the compensator of a counting process.