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.