A new proof of Kellerer’s theorem

Francis Hirsch; Bernard Roynette

ESAIM: Probability and Statistics (2012)

  • Volume: 16, page 48-60
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Hirsch, Francis, and Roynette, Bernard. "A new proof of Kellerer’s theorem." ESAIM: Probability and Statistics 16 (2012): 48-60. <http://eudml.org/doc/274380>.

@article{Hirsch2012,
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.},
author = {Hirsch, Francis, Roynette, Bernard},
journal = {ESAIM: Probability and Statistics},
keywords = {convex order; 1-martingale; peacock; Fokker-Planck equation; Kellerer's theorem; associated processes, 1-martingale},
language = {eng},
pages = {48-60},
publisher = {EDP-Sciences},
title = {A new proof of Kellerer’s theorem},
url = {http://eudml.org/doc/274380},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Hirsch, Francis
AU - Roynette, Bernard
TI - A new proof of Kellerer’s theorem
JO - ESAIM: Probability and Statistics
PY - 2012
PB - EDP-Sciences
VL - 16
SP - 48
EP - 60
AB - 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.
LA - eng
KW - convex order; 1-martingale; peacock; Fokker-Planck equation; Kellerer's theorem; associated processes, 1-martingale
UR - http://eudml.org/doc/274380
ER -

References

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  1. [1] C. Dellacherie and P.-A. Meyer, Probabilités et potentiel, Chapitres V à VIII, Théorie des martingales. Hermann (1980). Zbl0464.60001MR566768
  2. [2] F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associated martingales, with explicit constructions, Bocconi & Springer Series 3 (2011). Zbl1227.60001MR2808243
  3. [3] H.G. Kellerer, Markov-komposition und eine anwendung auf martingale. Math. Ann.198 (1972) 99–122. Zbl0229.60049MR356250
  4. [4] G. Lowther, Fitting martingales to given marginals. http://arxiv.org/abs/0808.2319v1 (2008). 

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