below: $$\label{hat-LTSR}
\hat{S}\left(z,t,t'\right):=\exp\left(
-\int_{t}^{t+t'}\exp\left(\hat{b}^{T}g^{-1}\left(
z,t,\tau\mid \hat{a}\right)\right)\hat{\lambda}_n^s
\left(\tau\right)d\tau\right),$$ where $\hat{a}$, $\hat{b}$ and $\hat{\lambda}_n^s$ are the estimators derived in section \[likelihood function\], which can be replaced by their penalized version as well. The consistency and asymptotic normality of the estimator is just a direct result of the theorem \[theorem-consistency\] and/or \[theorem-ALASSO\]. It is worthwhile to mention that is not possible to be constructed from the maximum partial likelihood estimators of the Cox model when temporal covariates are included. Because it is clear from that $\hat{S}$ relies on the information of the temporal covariates $Z$ within the forecast interval $\left[t,t+t'\right)$, which is not available from the maximum partial likelihood estimators.

Semi-martingale Longitudinal Processes
--------------------------------------

Although in the current discussion, the longitudinal process is assumed to have bounded variation and absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}_+$, the same framework should be extensible to more general cases where the longitudinal process may not have bounded variation (for example, given by a semi-martingale process). In a series of related works, the authors construct an explicit expression of the joint pdf of the event time and a semi-martingale longitudinal process, which enables us to construct the full information likelihood function. But the challenges 