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

Abstract

Let X be a point process and let X denote the filtration generated by X. In this paper we study martingale representation theorems in the filtration G obtained as an initial and progressive enlargement of the filtration X. In particular, the progressive enlargement is done by means of a whole point process H. We work here in full generality, without requiring any further assumption on the process H and we recover the special case in which X is enlarged progressively by a random time τ.

Martingale representations in progressive enlargement by multivariate point processes

“…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.…”

“…Remark 4.8. Point (i) of Theorem 3.5 in [12] is a particular case of previous theorem. More precisely in [12] the authors deal with a pair of non explosive point processes, a particular case of our setting, that is when d = 2, E 1 = E 2 = {1} and hence E \ {0, 0} = {(0, 1), (1, 0), (1, 1)}.…”

“…Point (i) of Theorem 3.5 in [12] is a particular case of previous theorem. More precisely in [12] the authors deal with a pair of non explosive point processes, a particular case of our setting, that is when d = 2, E 1 = E 2 = {1} and hence E \ {0, 0} = {(0, 1), (1, 0), (1, 1)}. They identify the two point processes with the jump measure of two different semimartingales, X and H. They call F the initially enlarged natural filtration of X and H the natural filtration of H and finally they indicate with G the right-continuous version of F ∨ H. They show that the semimartingale (X, H) has the G-WRP.…”

“…Some authors discuss the case when H is the natural filtration of a semimartingale (see e.g. 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.…”

“…In Section 5 we study the particular case when all explosion times are almost surely infinite and the mark's spaces are discrete. We devote Remark 4.8 and Remark 5.11 to the comparison with [12].…”