Title: From Azéma supermartingales of finite honest times to optional semimartingales of class-($Σ$),
Abstract: Given a finite honest time, we derive representations for the additive and
multiplicative decomposition of it's Azéma supermartingale in terms of
optional supermartingales and its running supremum. We then extend the notion
of semimartingales of class-$(\Sigma)$ to optional semimartingales with jumps
in its finite variation part, allowing one to establish formulas similar to the
Madan-Roynette-Yor option pricing formulas for larger class of processes.
Finally, we introduce the optional multiplicative systems associated with
positive submartingales and apply them to construct random times with given
Azéma supermartingale.