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

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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.

DOI : 10.1051/ps/2011164
Classification : 60E15, 60G44, 60G48, 60H10, 35K15
Keywords: 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},
     journal = {ESAIM: Probability and Statistics},
     pages = {48--60},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2011164},
     mrnumber = {2911021},
     zbl = {1277.60041},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2011164/}
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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

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