Peacocks and Associated Martingales, with Explicit Constructions

1. Front Matter

   Pages i-xxxi

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2. Some Examples of Peacocks

   • Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

   Pages 1-86

3. The Sheet Method

   • Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

   Pages 87-136

4. The Time Reversal Method

   • Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

   Pages 137-162

5. The Time Inversion Method

   • Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

   Pages 163-179

6. The Sato Process Method

   • Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

   Pages 181-221

7. The Stochastic Differential Equation Method

   • Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

   Pages 223-264

8. The Skorokhod Embedding (SE) Method

   • Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

   Pages 265-336

9. Comparison of Multidimensional Marginals

   • Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

   Pages 337-357

10. Back Matter

    Pages 359-385

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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.

• Brownian motion
• Convex order
• Markov processes
• Martingales
• Peacocks
• quantitative finance

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)

• Laboratoire d’Analyse et Probabilités, Université d’Évry-Val d’Essonne, France

  Francis Hirsch

• Institut Élie Cartan, Université Henri Poincaré, Nancy

  Christophe Profeta, Bernard Roynette

• Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris

  Marc Yor

• Institut Universitaire de France, France

  Marc Yor

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