Martingale representations in dynamic enlargement setting: the role of the accessible jump times

Abstract

Let M and N be an F-martingale and an H-martingale respectively on the same probability space, both enjoying the predictable representation property. We discuss how, under the assumption of the existence of an equivalent decoupling measure for F and H, the nature of the jump times of M and N affects the representation of the FVH-martingales. More precisely we show that the multiplicity of FVH depends on the behavior of the common accessible jump times of the two martingales. Then we propose an extension of Kusuoka's representation theorem to the case when the Brownian Motion is replaced by a semi-martingale which may jump at the default time with positive probability