Before revision: 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.
After revision: Given a finite honest time, we first derive a representation of the additive (the multiplicative ) decomposition of its Azema optional supermartingale in terms of the draw-down (the relative draw-down) of some optional supermartingales with continuous running supremum.