  We show that all local martingales with respect to the initially enlarged
natural filtration of a vector of multivariate point processes can be weakly
represented up to the minimum among the explosion times of the components. We
also prove that a strong representation holds if any multivariate point process
of the vector has almost surely infinite explosion time and discrete mark's
space. Then we provide a condition under which the components of the
multidimensional local martingale driving the strong representation are
pairwise orthogonal.
