On Azéma-Yor processes, their optimal properties and the Bachelier-drawdown equation
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 t...
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| Format: | Journal article |
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2009
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| _version_ | 1826306662218072064 |
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| author | Carraro, L Karoui, N Obloj, J |
| author_facet | Carraro, L Karoui, N Obloj, J |
| author_sort | Carraro, L |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T06:51:22Z |
| format | Journal article |
| id | oxford-uuid:fcaabcab-e63e-4c39-85b1-05025669086b |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:51:22Z |
| publishDate | 2009 |
| record_format | dspace |
| spelling | oxford-uuid:fcaabcab-e63e-4c39-85b1-05025669086b2022-03-27T13:22:38ZOn Azéma-Yor processes, their optimal properties and the Bachelier-drawdown equationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fcaabcab-e63e-4c39-85b1-05025669086bSymplectic Elements at Oxford2009Carraro, LKaroui, NObloj, JWe 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. |
| spellingShingle | Carraro, L Karoui, N Obloj, J On Azéma-Yor processes, their optimal properties and the Bachelier-drawdown equation |
| title | On Azéma-Yor processes, their optimal properties and the
Bachelier-drawdown equation |
| title_full | On Azéma-Yor processes, their optimal properties and the
Bachelier-drawdown equation |
| title_fullStr | On Azéma-Yor processes, their optimal properties and the
Bachelier-drawdown equation |
| title_full_unstemmed | On Azéma-Yor processes, their optimal properties and the
Bachelier-drawdown equation |
| title_short | On Azéma-Yor processes, their optimal properties and the
Bachelier-drawdown equation |
| title_sort | on azema yor processes their optimal properties and the bachelier drawdown equation |
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