A new proof of Kellerer's theorem
ESAIM: Probability and Statistics, Volume 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
Keywords: convex order, 1-martingale, peacock, Fokker-Planck equation
     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://archive.numdam.org/articles/10.1051/ps/2011164/}
AU  - Hirsch, Francis
AU  - Roynette, Bernard
TI  - A new proof of Kellerer's theorem
JO  - ESAIM: Probability and Statistics
PY  - 2012
SP  - 48
EP  - 60
VL  - 16
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2011164/
DO  - 10.1051/ps/2011164
LA  - en
ID  - PS_2012__16__48_0
ER  - 
%0 Journal Article
%A Hirsch, Francis
%A Roynette, Bernard
%T A new proof of Kellerer's theorem
%J ESAIM: Probability and Statistics
%D 2012
%P 48-60
%V 16
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2011164/
%R 10.1051/ps/2011164
%G en
%F PS_2012__16__48_0
Hirsch, Francis; Roynette, Bernard. A new proof of Kellerer's theorem. ESAIM: Probability and Statistics, Volume 16 (2012), pp. 48-60. doi : 10.1051/ps/2011164. http://archive.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).

Cited by Sources: