Carthaginian enlargement of filtrations

Abstract

This work is concerned with the theory of initial and progressive enlargements of a reference filtration F with a random time τ . We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alternative proofs to results concerning canonical decomposition of an F-martingale in the enlarged filtrations. Also, we address martingales' characterization in the enlarged filtrations in terms of martingales in the reference filtration, as well as predictable representation theorems in the enlarged filtrations.Keywords: initial and progressive enlargements of filtrations, predictable projection, canonical decomposition of semimartingales, predictable representation theorem.

An enlargement of filtration formula with applications to multiple non-ordered default times

“…Notice that a computational formula for Y g is already available in Lemma 5.13, however this formula is not adapted to the computation that we will do in this subsection. From Theorem 2.5 in [17], there exists a 6) where the predictable bracket M, M is calculated in the filtration F. The process u M (x) is known to satisfy…”

“…The projection type arguments have been previously used in Jeanblanc and Le Cam [18], Callegaro et al [6] and Kchia et al [25], to study the relationship between the filtrations F ⊂ F τ ⊂ F σ(τ ) , where F τ (resp. F σ(τ ) ) is the progressive (resp.…”

Information uncertainty related to marked random times and optimal investment

“…In particular, similarly to Jeulin (1980) (see also Callegaro et al (2013)), a stochastic process Z is G-adapted if and only if it can be written in the form…”

Dynamics of multivariate default system in random environment

“…We first recall a martingale criterion in the initial enlargement of filtration in Amendinger [3] (see also Callegaro, Jeanblanc and Zargari [7]). It corresponds in our setting to the total information filtration H = (H t ) t≥0 , H t = ∩ s>t (F s ∨ σ(χ)).…”