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We call peacock an integrable process which is increasing in the convex order; such a notion plays an important role in Mathematical Finance. A deep theorem due to Kellerer states that a process is a peacock if and only if it has the same one-dimensional marginals as a martingale. Such a martingale is then said to be associated to this peacock.In this monograph, we exhibit numerous examples of peacocks and associated martingales with the help of different methods: construction of sheets, time reversal, time inversion, self-decomposability, SDE, Skorokhod embeddings. They are developed in eight chapters, with about a hundred of exercises.
List of contents
"Some Examples of Peacocks."
"The Sheet Method."
"The Time Reversal Method."
"The Time Inversion Method."
"The Sato Process Method."
"The Stochastic Differential Equation Method."
The Skorokhod Embedding (SE) Method.
Comparison of Multidimensional Marginals.
Summary
We call peacock an integrable process which is increasing in the convex order; such a notion plays an important role in Mathematical Finance. A deep theorem due to Kellerer states that a process is a peacock if and only if it has the same one-dimensional marginals as a martingale. Such a martingale is then said to be associated to this peacock.
In this monograph, we exhibit numerous examples of peacocks and associated martingales with the help of different methods: construction of sheets, time reversal, time inversion, self-decomposability, SDE, Skorokhod embeddings. They are developed in eight chapters, with about a hundred of exercises.
Additional text
From the reviews:
“The authors provide several characterizations of the peacock property, and then continue to provide many different examples of peacocks. … For researchers, the book is a great opportunity to get introduced to this relatively new and fascinating branch of probability theory. For practitioners who want to create models for empirically given marginals (given, e.g., via option prices), the monograph should be a very valuable reference.” (Nicolas Perkowski, Zentralblatt MATH, Vol. 1227, 2012)
Report
From the reviews:
"The authors provide several characterizations of the peacock property, and then continue to provide many different examples of peacocks. ... For researchers, the book is a great opportunity to get introduced to this relatively new and fascinating branch of probability theory. For practitioners who want to create models for empirically given marginals (given, e.g., via option prices), the monograph should be a very valuable reference." (Nicolas Perkowski, Zentralblatt MATH, Vol. 1227, 2012)
