= Solution
Keep the <counting process> and <at-risk process> notation from the introductory derivation. Because the individual hazards add, the aggregate conditional intensity is
$$
\lambda(t)=\sum_iY_i(t)h_{0i}(t)+Y(t)h_1(t).
$$
The known individual contributions must be averaged over the current <risk set>, rather than over the original cohort. Define
$$
\overline h_0(t)=\frac{\sum_iY_i(t)h_{0i}(t)}{Y(t)}\qquad(Y(t)>0).
$$
Then $dN(t)/Y(t)$ estimates $\{\overline h_0(t)+h_1(t)\}dt$. Subtract the known average contribution to obtain the <offset-adjusted cumulative hazard estimator>
$$
\boxed{\widehat H_1(t)
=\sum_{a_j\leq t}\frac1{r_j}
-\int_0^t\mathbf1_{\{Y(u)>0\}}\frac{\sum_iY_i(u)h_{0i}(u)}{Y(u)}\,du.}
$$
Take the quotient to be zero when $Y=0$. If $M(t)=N(t)-\int_0^t\lambda(u)du$, direct substitution gives
$$
\widehat H_1(t)
=\int_0^t\mathbf1_{\{Y(u)>0\}}h_1(u)du
+\int_0^t\frac{\mathbf1_{\{Y(u)>0\}}}{Y(u)}\,dM(u).
$$
This establishes the target and the mean-zero estimation noise, under the usual integrability conditions. It estimates $H_1(t)$ while there is a nonempty <risk set>; no data identify its continuation after follow-up has ended. With common known $h_{0i}=h_0$ and individuals at risk throughout $[0,t]$, it reduces to $\widehat H_1(t)=\widehat H(t)-H_0(t)$.
The estimate may decrease between events because the known offset is subtracted continuously. That is not an algebraic error: it is an unconstrained estimating-equation estimator, not automatically a nonnegative monotone cumulative hazard estimate. If such shape constraints are imposed, the fitting procedure must explicitly account for them. The working hazard model itself must satisfy $h_{0i}+h_1\geq0$.
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