On Azéma-Yor processes, their optimal properties and the Bachelier-drawdown equation
Résumé
We study the class of Azema-Yor processes defined from a general semi-martingale with a continuous running maximum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the drawdown equation. Solutions of the latter have the drawdown property: they always stay above a given function of their past maximum. We then show that any process which satisfies the drawdown property is in fact an Azema-Yor process. The proofs exploit group structure of the set of Azema-Yor processes, indexed by functions, which we introduce.
We investigate in detail Azema-Yor martingales defined from a nonnegative local martingale converging to zero at infinity. We establish relations between average value at risk, drawdown function, Hardy-Littlewood transform and its inverse. In particular, we construct Azema-Yor martingales with a given terminal law and this allows us to rediscover the Azema-Yor solution to the Skorokhod embedding problem. Finally, we characterize Azema-Yor martingales showing they are optimal relative to the concave ordering of terminal variables among martingales whose maximum dominates stochastically a given benchmark.