Single Jump Filtrations: Preservation of the Local Martingale Property with Respect to the Filtration Generated by the Local Martingale

Abstract

Let M be a local martingale with respect to a so-called single jump filtration \(\mathbb {F}=\mathbb {F}(\gamma ,\mathcal {F})\) generated by a random time γ on a probability space \((\varOmega ,\mathcal {F},\mathsf P)\). It was recently mentioned by Herdegen and Herrmann (2016) that M is also a local martingale with respect to the filtration \(\mathbb {H}=\mathbb {F}^M\) that it generates if \(\mathcal {F}\) is the smallest σ-field with respect to which γ is measurable. We provide an example of a local martingale with respect to a general single jump filtration which is not a local martingale with respect to \(\mathbb {H}\). Then, we find necessary and sufficient condition for preserving the local martingale property with respect to \(\mathbb {H}\). The main idea of our constructions and the proofs is that \(\mathbb {H}\) is also a single jump filtration generated, in general, by other random time and σ-field. Finally, we prove that every σ-martingale in considered models is still a σ-martingale with respect to the filtration that it generates.