Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T16:10:29.520Z Has data issue: false hasContentIssue false
  • English
  • Français

On ℝd-valued peacocks

Published online by Cambridge University Press:  21 May 2013

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. francis.hirsch@univ-evry.fr
Bernard Roynette
Affiliation:
Institut Elie Cartan, Université Henri Poincaré, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France; bernard.roynette@iecn.u-nancy.fr
Get access

Abstract

In this paper, we consider ℝd-valued integrable processes which are increasing in the convex order, i.e. ℝd-valued peacocks in our terminology. After the presentation of some examples, we show that an ℝd-valued process is a peacock if and only if it has the same one-dimensional marginals as an ℝd-valued martingale. This extends former results, obtained notably by Strassen [Ann. Math. Stat. 36 (1965) 423–439], Doob [J. Funct. Anal. 2 (1968) 207–225] and Kellerer [Math. Ann. 198 (1972) 99–122].

Keywords

Convex ordermartingale1-martingalepeacock
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 444 - 454
Copyright
© EDP Sciences, SMAI, 2013

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

Cartier, P., Fell, J.M.G. and Meyer, P.-A., Comparaison des mesures portées par un convexe compact. Bull. Soc. Math. France 92 (1964) 435–445. Google Scholar
C. Dellacherie and P.-A. Meyer, Probabilités et potentiel, in Théorie des martingales, Chapitres V à VIII. Hermann (1980).
Doob, J.L., Generalized sweeping-out and probability. J. Funct. Anal. 2 (1968) 207–225. Google Scholar
Hirsch, F. and Roynette, B., A new proof of Kellerer’s theorem. ESAIM: PS 16 (2012) 48–60. Google Scholar
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).
Strassen, V., The existence of probability measures with given marginals. Ann. Math. Stat. 36 (1965) 423–439. Google Scholar