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.
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.
R
Azema-Yor
Az\'ema-Yor
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supremum
maximum
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Drawdown
drawdown
R
supremum
maximum
R
Azema-Yor
Az\'ema-Yor
R
Azema-Yor
Az\'ema-Yor
R
Secondly we study in detail Azema-Yor
We investigate in detail Az\'ema-Yor
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non-negative
nonnegative
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Average Value at Risk, Drawdown
average value at risk, drawdown
D
generalised
R
Azema-Yor
Az\'ema-Yor
R
Azema-Yor
Az\'ema-Yor
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characterise Azema-Yor
characterize Az\'ema-Yor