Martingale representation in progressive enlargement by the reference filtration of a semimartingale: a note on the multidimensional case

TL;DR

This paper extends martingale representation results to the multidimensional case under progressive filtration enlargement, with applications in credit risk modeling and the analysis of the multiplicity of combined filtrations.

Contribution

It introduces a new martingale representation theorem for the union of filtrations and applies it to determine the maximal multiplicity and to credit risk models involving default times.

Findings

Established a martingale representation under combined filtrations.

Identified the maximum multiplicity of combined reference filtrations.

Provided a new proof of Kusuoka's theorem for multidimensional semimartingales.

Abstract

Let X and Y be an m-dimensional F-semimartingale and an n-dimensional H-semimartingale respectively on the same probability space, both enjoying the strong predictable representation property. We propose a martingale representation result under the probability measure P for the square integrable G-martingales, where G is the union of F and H. As a first application we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of the union of F1,..., Fd, where, fixed i in (1,...,d), Fi is the reference filtration of a real martingale Mi, which enjoys the Fi-predictable representation property. A second application falls into the framework of credit risk modeling and in particular into the study of the progressive enlargement of the market filtration by a default time. More precisely, when the risky asset price is a multidimensional semimartingale enjoying the strong predictable representation property and the default time satisfies the density hypothesis, we present a new proof of the analogous of the classical theorem of Kusuoka.