I consider $(T_1,…, T_n)$ a random vector where $T_i$ is initially a point process associated to $N_t$ the counting process. We know that $N_t$ is a time inhomogeneous Poisson process with intensity function $\lambda(t)$, that is $N_t- N_s$ follows a Poisson($\int_{s}^{t}\lambda(s)ds$).
The fact $T_i$ is the point process associated to $N_t$ gives us that, for $A_k = [t_k, t_{k+h}]$ and $h$ sufficiently small :
$$ \cap_{k=1}^{n}\{T_k\in A_k\} = \cap_{k=1}^{n}\{N_{t_k+h}-N_{t_k} = 1\} $$
From this, using the indépendance of the increment of $N_t$ we get
$$ \mathbb{P}(\cap_{k=1} \{T_k\in A_k\}) =\mathbb{P}(\cap_{k=1}^{n}\{N_{t_k+h}-N_{t_k} = 1\}) = \Pi_{k=1}^{n}\left(e^{-\int_{t_k}^{t_{k + h}}\lambda(s)ds} \int_{t_k}^{t_{k + h}}\lambda(s)ds\right) = e^{-\sum_{k=1}^{n}\int_{t_k}^{t_{k + h}}\lambda(s)ds}\Pi_{k=1}^{n}\left(\int_{t_k}^{t_{k + h}}\lambda(s)ds\right) $$
From there, I don’t see how to pursue, I know that in the one dimensional case I need to take the derivative at some point to get the density, but here I do not see how to apply this ? Thus, I would like some hint (rather than complete answer) in order to finish this please.
Thank you