I encounter a problem in practical application but don't know where to start.
Suppose there is a time-series dataset $X_i(t_{i,j})$ of $n$ series, for $i = 1, 2, \dots, n$. For each time series $i$, we have a series of observations at different times $t_{i,j}$, $j = 1, 2, \dots, m_i$, where $(t_{i,j+1} - t_{i,j})$ is a random variable, i.e. not a constant.
The closest one that comes to my mind is the non-homogeneous Poisson process and its extension, but I can't pin this stochastic modeling problem to any specific stochastic process. Is it perhaps the jump process?