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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]
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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

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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).