  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.
