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

Abstract

ABSTRACTLet and be an m-dimensional -semi-martingale and an n-dimensional -semi-martingale, respectively, on the same probability space , both enjoying the predictable representation property. We propose two representation results for the square-integrable-martingales, where . As a first application, we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of , where, fixed , is the reference filtration of a martingale , which enjoys the -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.