From Az\'ema supermartingales of finite honest times to optional semimartingales of class-$(\Sigma)$}

Given a finite honest time, we derive representations for the additive and multiplicative decomposition of it's Az\'ema supermartingale in terms of optional supermartingales and its running supremum. We extend the notion of... [ view full abstract ]

Given a finite honest time, we derive representations for the additive and multiplicative decomposition of it's Az\'ema supermartingale in terms of optional supermartingales and its running supremum. We 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.