Martingale representations in progressive enlargement by multivariate point processes

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.