An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint

Henry-Labordère, Pierre; Tan, Xiaolu; Touzi, Nizar

doi:10.1016/j.spa.2016.03.003

Cited by 53 publications

(71 citation statements)

References 69 publications

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“…A multi-step coupling quite different from ours can be obtained by composing in a Markovian fashion the Left-Curtain transport kernels from µ t−1 to µ t , 1 ≤ t ≤ n, as discussed in [24]. In [32] the continuous-time limits of such couplings for n → ∞ are studied to find solutions of the so-called Peacock problem [25] where the marginals for a continuous-time martingale are prescribed; see also [23] and [33] for other continuous-time results with full marginal constraint. Early contributions related to the continuous-time martingale transport problem include [16,17,19,36,39,42].…”

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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“…Loosely speaking, the peacock problem is to give constructions of such martingales. Often such constructions are based on Skorokhod embedding or particular martingale transport plans, and often one is further interested in producing solutions with some additional optimality properties; see for example the recent works [29,40,41,35]. Given the intricacies of multi-period martingale optimal transport and Skorokhod embedding, it is necessary to make additional assumptions on the underlying marginals and desired optimality properties are in general not preserved in a straight forward way during the inherent limiting/pasting procedure.…”

Section: • Construction Of Peacocksmentioning

confidence: 99%

The geometry of multi-marginal Skorokhod Embedding

Beiglböck

Cox

Huesmann

2019

Probab. Theory Relat. Fields

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The Skorokhod Embedding Problem (SEP) is one of the classical problems in the study of stochastic processes, with applications in many different fields (cf. the surveys [48,34]). Many of these applications have natural multi-marginal extensions leading to the (optimal) multi-marginal Skorokhod problem (MSEP). Some of the first papers to consider this problem are [32,10,44]. However, this turns out to be difficult using existing techniques: only recently a complete solution was be obtained in [14] establishing an extension of the Root construction, while other instances are only partially answered or remain wide open.In this paper, we extend the theory developed in [2] to the multi-marginal setup which is comparable to the extension of the optimal transport problem to the multi-marginal optimal transport problem. As for the one-marginal case, this viewpoint turns out to be very powerful. In particular, we are able to show that all classical optimal embeddings have natural multi-marginal counterparts. Notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases.Moreover, our results also have consequences for the study of the martingale transport problem as well as the peacock problem.

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“…The literature on these problems can be traced back to Gyöngy [13], Dupire [9], Madan and Yor [27], the literature on fake Brownian motions (Hamza and Klebaner [14], Albin [2], Oleszkiewicz [29]), Pcocs (Hirsch et al [18]) and most recently and most relevantly for this paper, Henri-Labordère et al [16] and Källblad et al [25]. In addition there are strong connections with the literatures on the Skorokhod embedding problem (Sep), see Ob lój [28] and Hobson [20] for surveys, martingale optimal transport (Mot, Beiglböck et al [5], Beiglböck and Juillet [6]), and robust hedging of options (Hobson [19,20]).…”

Section: The Problemmentioning

confidence: 99%

Mimicking martingales

Hobson¹

2016

Ann. Appl. Probab.

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Given the univariate marginals of a real-valued, continuous-time martingale, (resp., a family of measures parameterised by t ∈ [0, T ] which is increasing in convex order, or a double continuum of call prices), we construct a family of pure-jump martingales which mimic that martingale (resp., are consistent with the family of measures, or call prices). As an example, we construct a fake Brownian motion. Then, under a further "dispersion" assumption, we construct the martingale which (within the family of martingales which are consistent with a given set of measures) has the smallest expected total variation. We also give a pathwise inequality, which in the mathematical finance context yields a model-independent sub-hedge for an exotic security with payoff equal to the total variation of the price process.1. The problem. In this article, we are concerned with the following problem:construct a fake version of M (or equivalently a process which mimics M), that is, construct [potentially on a new filtered probability space ( , F, F, P)] a stochastic process X = (X t ) 0≤t≤T such that X is a F-martingale and the univariate marginals of X are the same as those of M, but such that the joint marginals are different.The problem can be reformulated in two further ways. First, instead of beginning with a martingale M, we can begin with a family of laws (μ t ) 0≤t≤T which are increasing in convex order. Then the aim is to construct a filtered probability space ( , F, F, P) and a F-martingale X = (X t ) 0≤t≤T on that space such that P(X t ≤ x) = μ t ((−∞, x]) for all t and all x. Then we say that X is consistent with the measures (μ t ) 0≤t≤T . Second, but closely related, instead of beginning with a process or a set of measures we can work in the setting of mathematical finance and start with a double continuum of European call prices {C(t, k); 0 ≤ t ≤ T , 0 ≤ k < ∞} which satisfy no-arbitrage conditions. [We assume that we are working with discounted prices, and then the no-arbitrage conditions are that: for each t, C(t, k) is a decreasing convex function with C (t, 0+) ≥ 1 and lim k↑∞ C(t, k) = 0; C(t, 0) is a positive constant, independent of t; and C(t, k) is nondecreasing in t, for all k.] Then the aim is to find a model which is consistent with the given call prices, that is, a filtered probability space and a martingale X

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An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint

Cited by 53 publications

References 69 publications

Multiperiod martingale transport

Multiperiod martingale transport

The geometry of multi-marginal Skorokhod Embedding

Mimicking martingales

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