On $\mathbb {R}^d$-valued peacocks

Hirsch, Francis; Roynette, Bernard

doi:10.1051/ps/2012009

ℝ^{d}

-valued peacocks

Hirsch, Francis ; Roynette, Bernard

ESAIM: Probability and Statistics, Tome 17 (2013), pp. 444-454.

Résumé

In this paper, we consider ℝ^d-valued integrable processes which are increasing in the convex order, i.e. ℝ^d-valued peacocks in our terminology. After the presentation of some examples, we show that an ℝ^d-valued process is a peacock if and only if it has the same one-dimensional marginals as an ℝ^d-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].

Zbl

DOI : 10.1051/ps/2012009

Classification : 60E15, 60G44, 60G15, 60G48
Mots-clés : convex order, martingale, 1-martingale, peacock

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     author = {Hirsch, Francis and Roynette, Bernard},
     title = {On $\mathbb {R}^d$-valued peacocks},
     journal = {ESAIM: Probability and Statistics},
     pages = {444--454},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2012009},
     zbl = {1291.60085},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2012009/}
}

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Hirsch, Francis; Roynette, Bernard. On $\mathbb {R}^d$-valued peacocks. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 444-454. doi : 10.1051/ps/2012009. http://archive.numdam.org/articles/10.1051/ps/2012009/

Bibliographie
Cité par

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[3] J.L. Doob, Generalized sweeping-out and probability. J. Funct. Anal. 2 (1968) 207-225. | MR | Zbl

[4] F. Hirsch and B. Roynette, A new proof of Kellerer's theorem. ESAIM: PS 16 (2012) 48-60. | Numdam | MR | Zbl

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

[6] H.G. Kellerer, Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198 (1972) 99-122. | MR | Zbl

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

[8] V. Strassen, The existence of probability measures with given marginals. Ann. Math. Stat. 36 (1965) 423-439. | MR | Zbl

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