A new proof of Kellerer's theorem
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 48-60.

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.

DOI : 10.1051/ps/2011164
Classification : 60E15, 60G44, 60G48, 60H10, 35K15
Mots-clés : convex order, 1-martingale, peacock, Fokker-Planck equation 
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     author = {Hirsch, Francis and Roynette, Bernard},
     title = {A new proof of {Kellerer's} theorem},
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     pages = {48--60},
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Hirsch, Francis; Roynette, Bernard. A new proof of Kellerer's theorem. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 48-60. doi : 10.1051/ps/2011164. http://www.numdam.org/articles/10.1051/ps/2011164/

[1] C. Dellacherie and P.-A. Meyer, Probabilités et potentiel, Chapitres V à VIII, Théorie des martingales. Hermann (1980). | MR | Zbl

[2] F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associated martingales, with explicit constructions, Bocconi & Springer Series 3 (2011). | MR | Zbl

[3] H.G. Kellerer, Markov-komposition und eine anwendung auf martingale. Math. Ann. 198 (1972) 99-122. | MR | Zbl

[4] G. Lowther, Fitting martingales to given marginals. http://arxiv.org/abs/0808.2319v1 (2008).

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