Martingale representations in progressive enlargement by the reference filtration of a semi-martingale: a note on the multidimensional case

Let X and Y be an m-dimensional F-semi-martingale and an n-dimensional H-semi-martingale, respectively, on the same probability space (Omega, F, P), both enjoying the predictable representation property. We propose two representation results for the square-integrable G-martingales, where G = F boolean OR H. As a first application, we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of Vi-1(d) F-i where, fixed i is an element of (1, ..., d), F-i is the reference filtration of a martingale M-i, which enjoys the (P, F-i)-predictable representation property. This result helps us to identify a basis of martingales for the Poisson filtration enlarged by a general random time. A second application falls into the framework of credit risk modelling and in particular into the study of progressive enlargement of the market filtration by a default time. We present a new proof of the analogous of classical Kusuoka's theorem, when the risky asset price is a multidimensional semi-martingale enjoying the predictable representation property and the default time satisfies the density hypothesis