While I think I have acquired the essence of the nature of local martingales, I am still a bit uncomfortable with the several definitions that I come across in the literature. For instance ( I am omitting the reference to the filtered probability space for convenience):
Definition 1: A process $X $ is a local martingale if $X^{\tau_n} - X_0 $ is a uniformly integrable martingale where $\{\tau_n: n \in \mathbb{N}\} $ is a localizing sequence. This is the one I have always used.
Definition 2 A process $X $ is a local martingale if there is a localizing sequence $\{\tau_n: n \in \mathbb{N}\} $ such that $X^{\tau_n} $ is a martingale with respect to ${\cal F}_{t\wedge \tau_n}. $ In this case the author (R. Durrett), defines $X^\tau_t = 0 $ on $\{\tau =0\}. $
Definition 3 Given a right-continuous filtration, an adapted cadlag process $X $ is a local martingale if there exists a localizing sequence $\{\tau_n: n \in \mathbb{N}\} $ such that $X_{t\wedge \tau_n}\cdot 1_{\{\tau_n > 0\}} $ is a uniformly integrable martingale.
Definition 4 A process $X $ is said to be a local martingale if it is locally a right-continuous martingale. That is, there is a localizing sequence $\{\tau_n: n \in \mathbb{N}\} $ such that $X^{\tau_n}\cdot 1_{\tau_n > t_0} $ are martingales.
Is there a reference where the equivalence (if they are indeed equivalent) of the different definitions is explained ?
Thank you.