Root to Kellerer

Beiglböck, Mathias; Huesmann, Martin; Stebegg, Florian

doi:10.1007/978-3-319-44465-9_1

Cited by 14 publications

(28 citation statements)

References 17 publications

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“…One-step martingale optimal transport problems can alternately be studied as optimal Skorokhod embedding problems with marginal constraints; cf. [2,3,6,7]. A multi-marginal extension [1] of [2] is in preparation at the time of writing and the authors have brought to our attention that it will offer a version of Theorem 1.1 in the Skorokhod picture, at least in the case where µ 0 is atomless and some further conditions are satisfied.…”

Section: Background and Related Literaturementioning

confidence: 99%

Multiperiod martingale transport

Nutz

Stebegg

Tan

2020

Stochastic Processes and their Applications

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Consider a multiperiod optimal transport problem where distributions µ 0 , . . . , µ n are prescribed and a transport corresponds to a scalar martingale X with marginals X t ∼ µ t . We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property of their support, as simultaneous optimizers for a class of bivariate transport cost functions with a Spence-Mirrlees property, and by an order-theoretic minimality property. Left-monotone transports are unique if µ 0 is atomless, but not in general. In the one-period case n = 1, these transports reduce to the Left-Curtain coupling of Beiglböck and Juillet. In the multiperiod case, the bivariate marginals for dates (0, t) are of Left-Curtain type, if and only if µ 0 , . . . , µ n have a specific order property. The general analysis of the transport problem also gives rise to a strong duality result and a description of its polar sets. Finally, we study a variant where the intermediate marginals µ 1 , . . . , µ n−1 are not prescribed.

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Section: Background and Related Literaturementioning

confidence: 99%

Multiperiod martingale transport

Nutz

Stebegg

Tan

2020

Stochastic Processes and their Applications

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“…In most of this literature (see e.g., [14,11,9,36,24,16,19,22,39] and the references therein) the martingales may or not be Markov. The papers by Lowther [35,34] on limits of diffusion processes for the finite dimensional convergence permitted some authors to refocus on the Markov setting (see, e.g., [3,20,23]), rediscovering Kellerer's work by the way. Lowther's proof consists in adapting the local volatility coefficient of a SDE without drift-as indicated by Dupire in his very influential note [11] on financial engineering-in order to match the marginals of (µ t ) t∈R mollified in time and space.…”

Section: Moreovermentioning

confidence: 99%

The Markov-quantile process attached to a family of marginals

Boubel¹,

Juillet²

2021

Journal De L’École Polytechnique — Mathématiques

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“…For surveys with examples of Lipschitz kernels and Lipschitz-Markov martin gales, one can refer to [9] or [1].…”

Section: Definition 5 (Lipschitz Kernelmentioning

confidence: 99%

“…Hence one can carefully check that at least (1), (2) or (3) is not correct for the choice (q, q ) = (q 1 , q 2 ). In fact if g t (q 1 ) = g t (q 2 ) the criterion is not satisfied in (1). If g t (q 1 ) = g t (q 2 ), it is not satisfied in (2).…”

Section: Quantile Martingales and Characterisation Of The Markov Propmentioning

confidence: 99%

Peacocks Parametrised by a Partially Ordered Set

Juillet¹

2016

Lecture Notes in Mathematics

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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 [7] by Hirsch, Profeta, Roynette and Yor (Problem 7a-7b: "Find other versions of Kellerer's Theorem"). Case b) has been answered positively by Hirsch and Roynette in [8] 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 [14] 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 (µ t) t∈T non-decreasing in the stochastic order. We conclude the paper with open problems.

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Root to Kellerer

Cited by 14 publications

References 17 publications

Multiperiod martingale transport

Multiperiod martingale transport

The Markov-quantile process attached to a family of marginals

Peacocks Parametrised by a Partially Ordered Set

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