Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T23:42:39.113Z Has data issue: false hasContentIssue false
  • English
  • Français

A new proof of Kellerer’s theorem

Published online by Cambridge University Press:  26 March 2012

Francis Hirsch  and
Bernard Roynette
Francis Hirsch
Affiliation:
Laboratoire d’Analyse et Probabilités, Université d’Évry, Val d’Essonne, Boulevard F. Mitterrand, 91025 Évry Cedex, France. [email protected]
Bernard Roynette
Affiliation:
Institut Elie Cartan, Université Henri Poincaré, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France; [email protected]
Get access

Abstract

In this paper, we present a new proof of the celebrated theorem of Kellerer, stating that every integrable process, which increases in the convex order, has the same one-dimensional marginals as a martingale. Our proof proceeds by approximations, and calls upon martingales constructed as solutions of stochastic differential equations. It relies on a uniqueness result, due to Pierre, for a Fokker-Planck equation.

Keywords

Convex order1-martingalepeacockFokker-Planck equation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 48 - 60
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

C. Dellacherie and P.-A. Meyer, Probabilités et potentiel, Chapitres V à VIII, Théorie des martingales. Hermann (1980).
F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associated martingales, with explicit constructions, Bocconi & Springer Series 3 (2011).
Kellerer, H.G., Markov-komposition und eine anwendung auf martingale. Math. Ann. 198 (1972) 99–122. Google Scholar
G. Lowther, Fitting martingales to given marginals. http://arxiv.org/abs/0808.2319v1 (2008).