Given a finite honest time, we derive representations for the additive and multiplicativedecomposition of it's Az\'ema  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.
Given a finite honest time, we first derive a representation of the additive (the multiplicative) decomposition of its Az\'ema optional supermartingale in terms of the draw-down (the relative draw-down) of some optional supermartingales with continuous running supremum. 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 formulae similar to the Madan-Roynette-Yor option pricing formula for larger class of processes.
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derive representations for the additive and multiplicativedecomposition of it's
first derive a representation of the additive (the multiplicative) decomposition of its
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optional
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optional supermartingales and its
the draw-down (the relative draw-down) of some optional supermartingales with continuous
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We then
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
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formulas
formulae
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formulas
formula