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

Abstract

Let X and Y be an m-dimensional F-semi-martingale and an n-dimensional H-semi-martingale respectively on the same probability space (Ω, F, P ), both enjoying the strong predictable representation property. We propose a martingale representation result for the square-integrable (P, G)-martingales, where G = F ∨ H. As a first application we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of d i=1 F i , where, fixed i ∈ (1, . . . , d), F i is the reference filtration of a real martingale M i , which enjoys the (P, F i ) 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 semi-martingale 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 Kusuoka's theorem.

On the Propagation of the Weak Representation Property in Independently Enlarged Filtrations: The General Case

“…We recall that, if X has the PRP with respect to F = F X , Y has the PRP with respect to H = F Y and if F X and F Y are independent (at least under an equivalent martingale measure), Calzolari and Torti showed in [6] (under some additional conditions as, in particular, the triviality of F X 0 and F Y 0 and the locally square integrability of X and Y ) that every S ∈ H 2 (G) can also be represented as…”

“…We recall that the WRP is a more general property than the PRP (see, e.g., [10,Theorem 13.14] or Lemma 3.9). In Calzolari and Torti [6], the results of [5] are extended to multidimensional local martingales. We show in Corollary 3.10 that the results obtained by [5,6] can be reformulated in terms of the WRP of the G-local martingale Z = (X , Y ).…”

Martingale representation in the enlargement of the filtration generated by a point process

“…In [4] it is also shown that the multiplicity of G is, in general, equal to three. In [5], the results of [4] are generalized to multidimensional martingales M and N.…”

“…The proof of (i) is complete. For (ii) we refer to the proof of [5,Proposition 3.4]. The proof of the lemma is complete.…”

Martingale representations in progressive enlargement by multivariate point processes

“…the papers of Xue [26], Kchia and Protter [21] and more recently Di Tella [24] and Di Tella and Jeanblanc [12]). In great generality in [4], [6], [5] we studied the case when H is the reference filtration, but not necessarily the natural one, of a semimartingale. Here, focusing on the enlargement by multivariate point processes, we extend the results obtained in [12] about martingale representations of the initially enlarged natural filtration of a point process progressively enlarged by the natural filtration of a different point process.…”