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  .
Given a finite honest time, we first derive representations of the additive and the multiplicative decomposition of its Azema optional supermartingale in terms of the drawdown and the relative drawdown 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. Then we extend the notion of semimartingales of class-(\Sigma) by allowing for jumps in its finite variation part of the semimartingale decomposition. This allows one to establish the Madan-Roynette-Yor option pricing formula for a larger class of processes , and we apply the extended formula to the construction of finite honest times .
R
a representation
representations
R
(the multiplicative )
and the multiplicative
R
Az\'ema
Azema
R
draw-down (the relative draw-down)
drawdown and the relative drawdown
R
Finally,
Then
R
to optional semimartingales with
by allowing for
R
, allowing
of the semimartingale decomposition. This allows
D
formulae similar to
A
a
A
, and we apply the extended formula to the construction of finite honest times