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Peacocks and Associated Martingales, with Explicit Constructions

verfasst von: Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

Verlag: Springer Milan

Buchreihe : B&SS — Bocconi & Springer Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

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Inhaltsverzeichnis

Frontmatter

1. Some Examples of Peacocks

Abstract

We exhibit several classes of processes which are increasing in the convex order. Among these, an important class consists of the arithmetic means of martingales (which are closely related with Asian options).

Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

2. The Sheet Method

Abstract

To some peacocks constructed from a Brownian motion, we associate a martingale defined with the help of the Brownian sheet. We then generalize this approach in two directions:

1)

We first replace the Brownian motion (resp. the Brownian sheet) by a Lévy process (resp. a Lévy sheet).

2)

We then replace the Brownian motion (resp. the Brownian sheet) by a Gaussian process (resp. a Gaussian sheet).

Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

3. The Time Reversal Method

Abstract

We associate to some F₂-type peacocks (see Definition 1.8) a martingale by using a time reversal method. Then, quite similarly as in Chapter 2, where we exhibit some peacocks and associated martingales defined from the Brownian sheet, we construct, thanks to time reversal arguments, a new family of peacocks and associated martingales.

Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

4. The Time Inversion Method

Abstract

Denote by (Λ_t, t ≥ 0) an integrable Lévy process, i.e. for any t ≥ 0, [¦Λ_t¦] < ∞. Then, (t Λ_(1/t), t > 0) is a martingale in its natural filtration. Martingales of this type appear as being naturally associated to F₁-type peacocks or peacocks defined from squared Bessel processes of dimension 0, or, more generally stable CSBP with index γ ∈] 1, 2]. We then generalize the preceding results of this chapter in Theorem 4.5, through a more abstract approach. Finally, we give examples of applications of that theorem.

Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

5. The Sato Process Method

Abstract

We study various peacocks defined from self-decomposable laws. We construct associated martingales from Sato processes or Sato sheets.

Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

6. The Stochastic Differential Equation Method

Abstract

To certain peacocks (X_t, t ≥ 0), we associate martingales (M_t, t ≥ 0) which solve stochastic differential equations (SDE’s) of the form (Z_t = ∫ ₀ ^t σ (s, Z_s)dB_s, t ≥ 0).

Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

7. The Skorokhod Embedding (SE) Method

Abstract

Several Skorokhod embeddings — which are presented in a table in the introduction of this chapter — allow to associate martingales to certain peacocks.

Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

8. Comparison of Multidimensional Marginals

Abstract

We compare, in a Gaussian setting, the multidimensional marginals of some peacocks and some 1-martingales which are associated to them via several methods.

Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

Backmatter

Titel: Peacocks and Associated Martingales, with Explicit Constructions
verfasst von: Francis Hirsch
Christophe Profeta
Bernard Roynette
Marc Yor
Copyright-Jahr: 2011
Verlag: Springer Milan
Electronic ISBN: 978-88-470-1908-9
Print ISBN: 978-88-470-1907-2
DOI: https://doi.org/10.1007/978-88-470-1908-9