= Solution
In the <counting-process intensity in survival analysis>, the <counting process> $N_i(t)$ jumps once if an observed event occurs; the <at-risk process> $Y_i(t)$ is one immediately before $t$ only while individual $i$ is still observed and event-free. Conditional on that information, the chance of an event in a short interval is $h_i(t)\,dt+o(dt)$ for an at-risk individual and zero otherwise. This is the meaning of
$$
\mathbb E(dN_i(t)\mid\mathcal H_{t-})=Y_i(t)h_i(t)\,dt.
$$
The history $\mathcal H_{t-}$ is the <filtration> generated by the observed event and censoring histories, entry information, and available <covariates> strictly before $t$. In particular, it determines the current <risk set> but does not reveal future event times. The right-hand side is the infinitesimal <compensator of a counting process>; the equality is interpreted as an intensity statement to first order in $dt$.
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