We study the class of Azema-Yor (AY) processes defined from a general semimartingale with a continuous running supremum 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 supremum. We then show that any process which satisfies the drawdown property is in fact an AY process. The proofs exploit group structure of the set of AY processes, indexed by functions, which we introduce. Further, we study in detail AY martingales defined from a non-negative local martingale converging to zero at infinity.  In particular, we construct AY martingales with a given terminal law and this allows us to rediscover the AY solution to the Skorokhod embedding problem. Finally, we prove new optimal properties of AY martingales relative to concave ordering of terminal laws of martingales .
We study the class of Azema-Yor processes defined from a general semimartingale with a continuous running supremum 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 supremum. 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. Secondly we study in detail Azema-Yor martingales defined from a non-negative local martingale converging to zero at infinity. We establish relations between Average Value at Risk, Drawdown function, Hardy-Littlewood transform and its generalised 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 characterise Azema-Yor martingales showing they are optimal relative to the concave ordering of terminal variables among martingales whose maximum dominates stochastically a given benchmark .
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(AY)
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AY
Azema-Yor
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AY
Azema-Yor
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Further,
Secondly
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AY
Azema-Yor
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We establish relations between Average Value at Risk, Drawdown function, Hardy-Littlewood transform and its generalised inverse.
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AY
Azema-Yor
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AY
Azema-Yor
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prove new optimal properties of AY martingales relative to
characterise Azema-Yor martingales showing they are optimal relative to the
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laws of martingales
variables among martingales whose maximum dominates stochastically a given benchmark