Peacocks parametrised by a partially ordered set. (English) Zbl 1370.60082
Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLVIII. Cham: Springer (ISBN 978-3-319-44464-2/pbk; 978-3-319-44465-9/ebook). Lecture Notes in Mathematics 2168. Séminaire de Probabilités, 13-32 (2016).
Summary: We indicate some counterexamples to the peacock problem for families of (a) real measures indexed by a partially ordered set or (b) vectorial measures indexed by a totally ordered set. This is a contribution to an open problem of the book [F. Hirsch et al., Peacocks and associated martingales, with explicit constructions. New York, NY: Springer (2011; Zbl 1227.60001)] (Problem 7a–7b: “Find other versions of Kellerer’s theorem”).
Case (b) has been answered positively by F. Hirsch and B. Roynette [ESAIM, Probab. Stat. 17, 444–454 (2013; Zbl 1291.60085)] but the question whether a “Markovian” Kellerer theorem hold remains open. We provide a negative answer for a stronger version: A “Lipschitz-Markovian” Kellerer theorem will not exist.
In case (a) a partial conclusion is that no Kellerer theorem in the sense of the original paper [H. G. Kellerer, Math. Ann. 198, 99–122 (1972; Zbl 0229.60049)] can be obtained with the mere assumption on the convex order. Nevertheless we provide a sufficient condition for having a Markovian associate martingale. The resulting process is inspired by the quantile process obtained by using the inverse cumulative distribution function of measures \((\mu_{t})_{t\in T}\) non-decreasing in the stochastic order.
We conclude the paper with open problems.
For the entire collection see [Zbl 1359.60007].
Case (b) has been answered positively by F. Hirsch and B. Roynette [ESAIM, Probab. Stat. 17, 444–454 (2013; Zbl 1291.60085)] but the question whether a “Markovian” Kellerer theorem hold remains open. We provide a negative answer for a stronger version: A “Lipschitz-Markovian” Kellerer theorem will not exist.
In case (a) a partial conclusion is that no Kellerer theorem in the sense of the original paper [H. G. Kellerer, Math. Ann. 198, 99–122 (1972; Zbl 0229.60049)] can be obtained with the mere assumption on the convex order. Nevertheless we provide a sufficient condition for having a Markovian associate martingale. The resulting process is inspired by the quantile process obtained by using the inverse cumulative distribution function of measures \((\mu_{t})_{t\in T}\) non-decreasing in the stochastic order.
We conclude the paper with open problems.
For the entire collection see [Zbl 1359.60007].
MSC:
| 60G42 | Martingales with discrete parameter |
| 60G44 | Martingales with continuous parameter |
| 60B10 | Convergence of probability measures |
| 28A33 | Spaces of measures, convergence of measures |
References:
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