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

P. 219–231.

Gushchin A. A., Zhunussova A.

Let M be a local martingale with respect to a so-called single jump
filtration F = F(г ,F ) generated by a random time г on a probability space
(Omega,F ,P). It was recently mentioned by Herdegen and Herrmann (2016) that M
is also a local martingale with respect to the filtration H = F^M that it generates
if F is the smallest у-field with respect to which г 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 H. Then, we find necessary and
sufficient condition for preserving the local martingale property with respect to H.
The main idea of our constructions and the proofs is that H is also a single jump
filtration generated, in general, by other random time and у-field. Finally, we prove
that every у-martingale in considered models is still a у-martingale with respect to
the filtration that it generates.

Keywords: filtration, local martingale, sigma-martingale, single jump filtration

In book

Operator Theory and Harmonic Analysis

* 2: Probability-Analytical Models, Methods and Applications. , Cham: Springer, 2021.