The strong predictable representation property in initially enlarged filtrations under the density hypothesis

Abstract

We study the strong predictable representation property in filtrations initially enlarged with a random variable L. We prove that the strong predictable representation property can always be transferred to the enlarged filtration as long as the classical density hypothesis of [Jac85] holds.This generalizes the existing martingale representation results and does not rely on the equivalence between the conditional and the unconditional laws of L. Depending on the behavior of the density process at zero, different forms of martingale representation are established. The results are illustrated in the context of hedging contingent claims under insider information.

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

“…In [10], Fontana proved that the PRP of a d-dimensional local martingale M with respect to F propagates to G, if G the initial enlargement of F by a random time τ which satisfies Jacod's absolutely continuity hypothesis. In particular, in [10] it is shown that the multiplicity of G is also d.…”

Martingale representations in progressive enlargement by multivariate point processes

“…Several authors deal with the topic of propagation of martingale representations from a fixed filtration F to an enlargement G. We list here some papers, without claiming to be exhaustive. As far as the case of initial enlargement is concerned we recall the contributions of Amendinger [1], Amendinger, Becherer and Schweizer [2], Grorud and Pontier [14] and the more recent paper of Fontana [13]. In the framework of progressive enlargement, motivated by the applications in credit risk modeling, most of the authors deal with the enlargement by the natural filtration of the occurrence process of a random time and among the others we mention the seminal paper of Kusuoka [22], the contributions of Jeanblanc and Le Cam [19], Jeanblanc and Song [19], Di Tella [11], Di Tella and Engelbert [25].…”

Filtration shrinkage, the structure of deflators, and failure of market completeness

“…The complementary problem of whether certain no-arbitrage and completeness conditions are preserved after filtration enlargement has been studied extensively, but is not considered in the present paper; we instead refer to Coculescu et al [11], Fontana et al [16], Jeanblanc and Song [19], Acciaio et al [1], Song [39], Aksamit et al [2], Song [40], Aksamit et al [3], Chau et al [9], Fontana [15], Chau et al [8] and the references therein. Filtration shrinkage and its effect on semimartingale characteristics has been studied by several researchers.…”