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摘要翻译：
研究了一类由一般半鞅定义的具有连续运行极大过程的Az\'ema-yor过程。我们证明了它们是Bachelier随机微分方程的唯一强解，并证明了该方程与降深方程是等价的。后者的解具有下降性质：它们总是保持在其过去最大值的给定函数之上。然后我们证明了任何满足降额性质的过程实际上都是一个AZ\'ema-yor过程。证明利用了我们所引入的以函数为索引的Az\'ema-yor过程集的群结构。本文详细地研究了由无穷远收敛到零的非负局部鞅定义的Az\'ema-yor鞅。建立了风险平均值、降额函数、Hardy-Littlewood变换及其逆函数之间的关系。特别地，我们构造了具有给定终端律的Az\'ema-yor鞅，这使得我们可以重新发现Skorokhod嵌入问题的Az\'ema-yor解。最后，我们刻划了Az\'ema-yor鞅，证明了它相对于最大值随机支配给定基准的鞅之间终端变量的凹序是最优的。
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英文标题：
《On Az\'ema-Yor processes, their optimal properties and the
Bachelier-drawdown equation》
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作者：
Laurent Carraro, Nicole El Karoui, Jan Ob{\l}\'oj
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最新提交年份：
2012
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分类信息：
一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance 数量金融学
二级分类：Pricing of Securities 证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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英文摘要：
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.
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PDF链接：
https://arxiv.org/pdf/0902.1328
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