The Wasserstein space of stochastic processes

Bartl, Daniel; Beiglböck, Mathias; Pammer, Gudmund

doi:10.48550/arxiv.2104.14245

Cited by 8 publications

(15 citation statements)

References 51 publications

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“…In the context of mathematical finance, the adapted Wasserstein distance, that is a variation of the Wasserstein distance, can be used to obtain sharp quantitative results for sequential decision making problems and in robust finance, see [11,17] among others. For µ, ν ∈ P p (R dT ), the adapted Wasserstein-p distance is defined by…”

Section: Optimal Transportmentioning

confidence: 99%

“…Example 10 (Adapted Wasserstein barycenters). Even though (P p (R dT ), AW p ) on its own is not a geodesic space, its completion (F P p , AW p ) is geodesically complete; see [17], which also gives a probabilistic interpretation of the aforementioned space as a Wasserstein space of stochastic processes. Moreover, (F P p , AW p ) provides a way of taking barycenters of stochastic processes that take into account path properties as well as the arrow of time encoded in the underlying filtrations.…”

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confidence: 99%

“…The existence of η can be shown along the lines of Example 9 due to the characterization of relative compact sets given in [17,Theorem 1.7]. ♦…”

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confidence: 99%

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Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

Acciaio¹,

Kratsios²,

Pammer³

2022

Preprint

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A. We introduce a universal class of geometric deep learning models, called metric hypertransformers (MHTs), capable of approximating any adapted map F : X Z → Y Z with approximable complexity, where X ⊆ R d and Y is any suitable metric space, and X Z (resp. Y Z ) capture all discrete-time paths on X (resp. Y ). Suitable spaces Y include various (adapted) Wasserstein spaces, all Fréchet spaces admitting a Schauder basis, and a variety of Riemannian manifolds arising from information geometry. Even in the static case, where f : X → Y is a Hölder map, our results provide the first (quantitative) universal approximation theorem compatible with any such X and Y . Our universal approximation theorems are quantitative, and they depend on the regularity of F, the choice of activation function, the metric entropy and diameter of X , and on the regularity of the compact set of paths whereon the approximation is performed. Our guiding examples originate from mathematical finance. Notably, the MHT models introduced here are able to approximate a broad range of stochastic processes' kernels, including solutions to SDEs, many processes with arbitrarily long memory, and functions mapping sequential data to sequences of forward rate curves.

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Section: Optimal Transportmentioning

confidence: 99%

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confidence: 99%

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Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

Acciaio¹,

Kratsios²,

Pammer³

2022

Preprint

Self Cite

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“…Theorem 1.1 goes beyond the above examples by capturing the effect of the interplay of convexity (including κ < 0) and support size on the contraction properties of the Brownian transport map. 2 The reason why we can prove the results in Theorem 1.1, which are unknown for the Brenier map, is because the Malliavin calculus available in the Wiener space allows us to write a differential equation for the derivative of the Brownian transport map, which in turn shows that it is a contraction. This feature does not have an analogue in optimal transport maps (but see [21, equation (1.8)] for a different transport map).…”

Section: Introductionmentioning

confidence: 99%

“…where the infimum is taken over all maps ξ : Ω → H 1 such that Law(ω + ξ(ω)) = µ with ω ∼ ν, and with the additional requirement that ξ is an adapted process. This notion of optimality, sometimes referred to as adapted optimal transport, has recently gained a lot of traction (e.g, [2] and references therein). The connection between these transport maps and the Brownian transport map follows from the work of Lassalle [26].…”

Section: Introductionmentioning

confidence: 99%

The Brownian transport map

Mikulincer¹,

Shenfeld²

2021

Preprint

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Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open.The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan-Lovász-Simonovits conjecture.We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors.1 The Wiener measure γ is such that Ω ∋ ω ∼ γ is a standard Brownian motion in R d where Ω is the classical Wiener space of continuous paths in R d parameterized by time t ∈ [0, 1].

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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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The Wasserstein space of stochastic processes

Cited by 8 publications

References 51 publications

Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

The Brownian transport map

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

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