Before revision:  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.
After revision:  The multiplicative representation then allows us to give a characterisation of finite honest times using a family of non-negative optional supermartingales with continuous running supremum which converges to zero at infinity. Finally, 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.