On Azéma–Yor processes, their optimal properties and the Bachelier–drawdown equation

Abstract

We study the class of Azéma–Yor processes defined from a general semimartingale 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 Azéma–Yor process. The proofs exploit group structure of the set of Azéma–Yor processes, indexed by functions, which we introduce.

We investigate in detail Azéma–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 Azéma–Yor martingales with a given terminal law and this allows us to rediscover the Azéma–Yor solution to the Skorokhod embedding problem. Finally, we characterize Azéma–Yor martingales showing they are optimal relative to the concave ordering of terminal variables among martingales whose maximum dominates stochastically a given benchmark.