On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation

TL;DR

This paper explores Azéma-Yor processes, demonstrating their unique solutions to the Bachelier and drawdown equations, and establishing their optimality and applications in stochastic embedding and risk management.

Contribution

It introduces a group structure for Azéma-Yor processes, characterizes their solutions, and connects them to optimal martingale problems and risk measures.

Findings

Azéma-Yor processes are unique solutions to the Bachelier and drawdown equations.

They can be constructed with prescribed terminal laws using the Skorokhod embedding.

Azéma-Yor martingales are optimal in concave order among certain martingales.

Abstract

We study the class of Az\'ema-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 Az\'ema-Yor process. The proofs exploit group structure of the set of Az\'ema-Yor processes, indexed by functions, which we introduce. We investigate in detail Az\'ema-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 Az\'ema-Yor martingales with a given terminal law and this allows us to rediscover the Az\'ema-Yor solution to the Skorokhod embedding problem. Finally, we characterize Az\'ema-Yor martingales showing they are optimal relative to the concave ordering of terminal variables among martingales whose maximum dominates stochastically a given benchmark.