On the Propagation of the Weak Representation Property in Independently Enlarged Filtrations: The General Case

Abstract

AbstractIn this paper, we investigate the propagation of the weak representation property (WRP) to an independently enlarged filtration. More precisely, we consider an $${\mathbb {F}}$$ F -semimartingale X possessing the WRP with respect to $${\mathbb {F}}$$ F and an $${\mathbb {H}}$$ H -semimartingale Y possessing the WRP with respect to $${\mathbb {H}}$$ H . Assuming that $${\mathbb {F}}$$ F and $${\mathbb {H}}$$ H are independent, we show that the $${\mathbb {G}}$$ G -semimartingale $$Z=(X,Y)$$ Z = ( X , Y ) has the WRP with respect to $${\mathbb {G}}$$ G , where $${\mathbb {G}}:={\mathbb {F}}\vee {\mathbb {H}}$$ G : = F ∨ H . In our setting, X and Y may have simultaneous jump-times. Furthermore, their jumps may charge the same predictable times. This generalizes all available results about the propagation of the WRP to independently enlarged filtrations.