Enlargement of filtration and predictable representation property for semi-martingales

Abstract

We present two examples of loss of the predictable representation property for semi-martingales by enlargement of the reference filtration. First of all we show that the predictable representation property for a semi-martingale X does not transfer from the reference filtration F to a larger filtration G if the information starts growing up to a positive time. Then we study the case G = F ∨ H when there exists a second special semi-martingale Y enjoying the predictable representation property with respect to H. We establish conditions under which the triplet (X, Y, [X, Y ]) enjoys the predictable representation property with respect to G.

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

“…Proof. i1) This statement is the extension to the multidimensional case of Lemma 4.2 in [6]. Thanks to point i) of Lemma 3.1 its proof is exactly the same.…”

“…When X and Y are not trivial semi-martingales, that is when they do not coincide with M and N respectively, under suitable assumptions, we are able to prove that the representation is the same. We stress that the approach is slightly different from that used to handle the unidimensional case (see Section 4.2 in [6]) and also that the hypotheses are simpler than those required in that paper. Here the question reduces to ask for conditions under which the (P X , F)-p.r.p.…”

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

“…In Section 3, we establish the main results of the present paper, Theorem 3.6 below, about the propagation of the WRP and the PRP. Section 4 is devoted to the study of the orthogonality of the fundamental martingales which are used for the martingale representation in G. In particular, as a consequence of Theorem 3.6, we generalize the results obtained in [4] on the martingale representation to the case of a non-trivial initial filtration in the context of point processes. In the appendix we discuss a lemma which will be useful to study the orthogonality of the involved martingales.…”

“…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.…”

“…Similarly, in [4], Calzolari and Torti consider two local martingales M and N possessing the PRP each in its own filtration and then they study the propagation of the PRP to G. The assumptions in [4] imply that M and N are independent and that G 0 is trivial. 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.…”