to extend the current work to the situation with the semi-martingale longitudinal measurements are the identification of the resulting model and the challenge in computation. For the identification issue, it is clear from the proof \[proof-identification\] that in the current framework, the identification relies on detailed analysis of the solution trajectories of ODE system induced by the function . In the case of semi-martingale longitudinal measurements, the ODE system will be replaced with a more complicated partial differential equation (PDE) system. Although it seems that there is no barrier to make the same trick in proof \[proof-identification\] invalid, the details to transplant the proof \[proof-identification\] to the semi-martingale case is open to future studies. In the aspect of computation, we have to apply numerical method to a PDE system in place of an ODE system, while,as known, the numerical method to solve PDE system is much more time consuming. A potential solution to the computation issue is to utilize the relation between PDE systems and the semi-martingale processes, through which simulating the underlying process could yield exactly the same solution to the PDE problem. The details of implementing that idea are left as another open problem for further study.

Proof 