model even if only the observations $\left\{\left(Z_{i,T_i},T_i\right):i=1,\dots,n\right\}$ are available, where all the observations $Z_{i,T_j}$ with $T_j<T_i$ are missing. Our procedure is based on joint modelling the longitudinal process that generates the time-dependent covariates and the time to event. The topic of joint model has been widely discussed ([@henderson2000joint; @song2002semiparametric; @hsieh2006joint; @ye2008semiparametric; @rizopoulos2011dynamic; @kim2013joint; @lawrence2015joint]). A comprehensive review is also available in [@tsiatis2004joint; @sousa2011review; @ibrahim2010basic]. However, all these works require the condition that the number of observations of the longitudinal measurement before the time to event is greater than the dimension of the longitudinal process, which restricts their usefulness for the input data with only one observation of the covariate value at the event time. In this paper, we propose an alternative specification of the joint model. Formally, we only assume that the longitudinal measurements follow a bounded variational process that can be expressed as a stochastic integral as below: $$\label{potential-growth-process}
\mathcal{Z}(t):=Z_{0}+\int_{0}^{t}Y(s)\epsilon(s)ds.$$ where $Z_{0}$ represents the initial value and $Y(t):=I\left(T>t\right)$. The model consists of two components:

\(1) the longitudinal process: $$\label{longitudinal}
Z(t)=Z_0+\int_{0}^{t}\epsilon(s)ds$$ which characterizes the evolution of the longitudinal measurements, we assume the conditional expectation of the increment rate $\epsilon(t)$ has the following parametric form $$\label{conditional-expectation-of-growth-rate}
q\left(z,t\mid a_{0}\right) := E\left(\epsilon(t)\mid Z(t)=z,a_{0}\right);$$ 