Optimal Couplings on Wiener Space and An Extension of Talagrand’s Transport Inequality

Föllmer, Hans

doi:10.1007/978-3-030-98519-6_7

Cited by 9 publications

(6 citation statements)

References 12 publications

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“…Continuous time adapted Wasserstein distance is used to derive continuity of optimal stopping in [1] and to derive stability properties of pricing, hedging and utility maximization in [3]. Remarkably, it satisfies Talagrand type inequalities with respect to specific entropy, see [17]. There is a rich literature on adapted versions of the Wasserstein distance in discrete time, see e.g.…”

Section: The Cn-transformation and Adapted Wasserstein Distance 31 (G...mentioning

confidence: 99%

Approximation of martingale couplings on the line in the adapted weak topology

Beiglböck

Jourdain

Margheriti

et al. 2022

Probab. Theory Relat. Fields

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Our main result is to establish stability of martingale couplings: suppose that $$\pi $$ π is a martingale coupling with marginals $$\mu , \nu $$ μ , ν . Then, given approximating marginal measures $$\tilde{\mu }\approx \mu , \tilde{\nu }\approx \nu $$ μ ~ ≈ μ , ν ~ ≈ ν in convex order, we show that there exists an approximating martingale coupling $$\tilde{\pi }\approx \pi $$ π ~ ≈ π with marginals $$\tilde{\mu }, \tilde{\nu }$$ μ ~ , ν ~ . In mathematical finance, prices of European call/put option yield information on the marginal measures of the arbitrage free pricing measures. The above result asserts that small variations of call/put prices lead only to small variations on the level of arbitrage free pricing measures. While these facts have been anticipated for some time, the actual proof requires somewhat intricate stability results for the adapted Wasserstein distance. Notably the result has consequences for several related problems. Specifically, it is relevant for numerical approximations, it leads to a new proof of the monotonicity principle of martingale optimal transport and it implies stability of weak martingale optimal transport as well as optimal Skorokhod embedding. On the mathematical finance side this yields continuity of the robust pricing problem for exotic options and VIX options with respect to market data. These applications will be detailed in two companion papers.

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Section: The Cn-transformation and Adapted Wasserstein Distance 31 (G...mentioning

confidence: 99%

Approximation of martingale couplings on the line in the adapted weak topology

Beiglböck

Jourdain

Margheriti

et al. 2022

Probab. Theory Relat. Fields

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“…Gantert [18,Kapitel II.4] shows that h is the rate function in a large deviations principle associated to a randomized Donsker-type approximation of Brownian motion. The specific relative entropy is also studied by Föllmer [16,15] who uses it to establish Talagrand-type inequalities on the Wiener space beyond the absolutely continuous case. In particular he proves that the squared adapted Wasserstein distance between a continuous martingale and Brownian motion is bounded from above by twice the specific relative entropy.…”

Section: On Specific Relative Entropymentioning

confidence: 99%

“…where Σ t stands for the density of the quadratic variation at time t for the canonical process under Q. More generally [18,16] show that…”

Section: On Specific Relative Entropymentioning

confidence: 99%

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Robust contracting in general contract spaces

2021

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In a sports game between two teams, the chance the home team wins is initially x 0 ∈ (0, 1) and finally 0 or 1. As an idealization we take a continuous time interval [0, 1] and consider the process M = (M t ) t∈[0,1] giving the probability at time t that the home team wins. This is a martingale which we idealize further to have continuous paths. We consider the problem to find the most random martingale M of this type, where 'most random' is interpreted as a maximal entropy criterion. We observe that this max-entropy win-martingale M also minimizes specific relative entropy with respect to Brownian motion in the sense of Gantert [18] and use this to prove that M is characterized by the stochastic differential equationTo derive the form of the optimizer we use a scaling argument together with a new first order condition for martingale optimal transport which may be of interest in its own right.

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“…It is also shown in [3] that * is the process whose evolution follows the movement of a Brownian particle as closely as possible with respect to an adapted Wasserstein distance (see e.g. [2,22]) subject to the marginal conditions 0 ∼ and 1 ∼ . These properties motivate to call the martingale * stretched Brownian motion between and as in [3].…”

Section: Benamou-brenier Transport Problem and Mccann Interpolation I...mentioning

confidence: 99%

Martingale Benamou–Brenier: A probabilistic perspective

Backhoff-Veraguas¹,

Beiglböck²,

Huesmann³

et al. 2020

Ann. Probab.

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A. In classical optimal transport, the contributions of Benamou-Brenier and Mc-Cann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas.Stretched Brownian motion provides an analogue for the martingale version of this problem. In this article we provide a characterization in terms of gradients of convex functions, similar to the characterization of optimizers in the classical transport problem for quadratic distance cost.

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Optimal Couplings on Wiener Space and An Extension of Talagrand’s Transport Inequality

Cited by 9 publications

References 12 publications

Approximation of martingale couplings on the line in the adapted weak topology

Approximation of martingale couplings on the line in the adapted weak topology

Robust contracting in general contract spaces

Martingale Benamou–Brenier: A probabilistic perspective

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