arXiv subjects

Stefan Schrott

Publications and source records attributed to Stefan Schrott.

9 recordsLinked to original sources

arXiv2026

Couplings Farthest from the Independent Gaussian

Motivated by Wasserstein measures of dependence, we study the largest possible 2-Wasserstein distance between a joint distribution and the product of its prescribed marginals. For two uniform marginals, Catalano and Lavenant conjectured that the monotone and antimonotone couplings maximize the distance from the independent coupling. We prove the Gaussian analogue for an arbitrary number $n\geq 2$ of one-dimensional standard Gaussian marginals. More generally, for every probability measure $μ$ on $\mathbb R$ with finite second moment, we characterize the laws on $\mathbb{R}^n$ with all marginals equal to $μ$ that are farthest from the $n$-dimensional standard Gaussian.

math.PR↗

arXiv2026

Delta-Epsilon-Common Knowledge and Quantitative Agreement Theorems

Aumann defined common knowledge mathematically and established his now famous Agreement Theorem. We present a novel approach to quantifying how close individuals are to commonly knowing events, $(δ,ε)$-common knowledge, which is defined for any (and not just countable) probability spaces, and provide quantitative versions of the key results in this field. Specifically, we do this for Aumann's Agreement Theorem and Nielsen's extension thereof to random variables, as well as for the setting in which posteriors are communicated back and forth between individuals. Our results apply in particular to noisy communication settings.

econ.TH↗

arXiv2025

A Brenier Theorem on $(P_2 (...P_2(H)...), W_2 )$ and Applications to Adapted Transport

We develop Brenier theorems on iterated Wasserstein spaces. For a separable Hilbert space $H$ and $N\geq 1$, we construct a full-support probability $Λ$ on $P_2^{N}(H)= P_2(... P_2(H)...)$ that is transport regular: for every $Q$ with finite second moment, transporting $Λ$ to $Q$ with cost $W_2^2$ admits a unique optimizer, and this optimizer is of Monge type. The analysis rests on a characterization of optimal couplings on $P_2(H)$ and, more generally, on $P_2^{N}(H)$ via convex potentials on the Lions lift; in the latter case we employ a new adapted version of the lift tailored to the $N$-step structure. A key idea is a new identification between optimal-transport $c$-conjugation (with $c$ given by maximal covariance) and classical convex conjugation on the lift. A primary motivation comes from the adapted Wasserstein distance $AW_2$: our results yield a first Brenier theorem for $AW_2$ and characterize $AW_2^2$-optimal couplings through convex functionals on the space of $L_2$-processes.

math.PR↗

arXiv2025

Denseness of biadapted Monge mappings

Adapted or causal transport theory aims to extend classical optimal transport from probability measures to stochastic processes. On a technical level, the novelty is to restrict to couplings which are bicausal, i.e. satisfy a property which reflects the temporal evolution of information in stochastic processes. We show that in the case of absolutely continuous marginals, the set of bicausal couplings is obtained precisely as the closure of the set of (bi-) adapted processes. That is, we obtain an analogue of the classical result on denseness of Monge couplings in the set of Kantorovich transport plans: bicausal transport plans represent the relaxation of adapted mappings in the same manner as Kantorovich transport plans are the appropriate relaxation of Monge-maps.

math.PR↗

arXiv2025

On Strassen's Theorem for support functions

Strassen established that there exists a two step martingale with marginal distributions $μ$, $ν$ if and only if $μ$, $ν$ are in convex order. Recently Choné-Gozlan-Kramarz obtained a transport characterization of the stochastic order defined by convex positively 1-homogeneous functions, in the spirit of Strassen's theorem under certain technical assumptions. In this note we prove the Choné-Gozlan-Kramarz result in full generality. We also observe that the restriction of the result to the case where $μ, ν$ are supported on a half space is equivalent to Strassen's classical theorem.

math.PR↗

arXiv2025

The Fundamental Theorem of Weak Optimal Transport

The fundamental theorem of classical optimal transport establishes strong duality and characterizes optimizers through a complementary slackness condition. Milestones such as Brenier's theorem and the Kantorovich-Rubinstein formula are direct consequences. In this paper, we generalize this result to non-linear cost functions, thereby establishing a fundamental theorem for the weak optimal transport problem introduced by Gozlan, Roberto, Samson, and Tetali. As applications we provide concise derivations of the Brenier--Strassen theorem, the convex Kantorovich--Rubinstein formula and the structure theorem of entropic optimal transport. We also extend Strassen's theorem in the direction of Gangbo--McCann's transport problem for convex costs. Moreover, we determine the optimizers for a new family of transport problems which contains the Brenier--Strassen, the martingale Benamou--Brenier and the entropic martingale transport problem as extreme cases.

math.PR↗

arXiv2025

The Wasserstein Space of Stochastic Processes in Continuous Time

Researchers from different areas have independently defined extensions of the usual weak convergence of laws of stochastic processes with the goal of adequately accounting for the flow of information. Natural approaches are convergence of the Aldous--Knight prediction process, Hellwig's information topology, convergence in adapted distribution in the sense of Hoover--Keisler and the weak topology induced by optimal stopping problems. The first main contribution of this article is that on continuous processes with natural filtrations there exists a canonical adapted weak topology which can be defined by all of these approaches; moreover, the adapted weak topology is metrized by a suitable adapted Wasserstein distance $\mathcal{AW}$. While the set of processes with natural filtrations is not complete, we establish that its completion consists precisely of the space ${\rm FP}$ of stochastic processes with general filtrations. We also show that $({\rm FP}, \mathcal{AW})$ exhibits several desirable properties. Specifically, it is Polish, martingales form a closed subset and approximation results such as Donsker's theorem extend to $\mathcal{AW}$.

math.PR↗

arXiv2024

A Probabilistic View on the Adapted Wasserstein Distance

Causal optimal transport and adapted Wasserstein distance have applications in different fields from optimization to mathematical finance and machine learning. The goal of this article is to provide equivalent formulations of these concepts in classic probabilistic language. In particular, we prove a Skorokhod representation theorem for adapted weak convergence, reformulate the equivalence of stochastic processes using Markovian lifts, and give an expression for the adapted Wasserstein distance based on representing processes on a common stochastic basis.

math.PR↗

arXiv2024

Representing General Stochastic Processes as Martingale Laws

Random variables $X^i$, $i=1,2$ are 'probabilistically equivalent' if they have the same law. Moreover, in any class of equivalent random variables it is easy to select canonical representatives. The corresponding questions are more involved for processes $X^i$ on filtered stochastic bases $(Ω^i, \mathcal F^i, \mathbb P^i, (\mathcal F^i_t)_{t\in [0,1]})$. Here equivalence in law does not capture relevant properties of processes such as the solutions to stochastic control or multistage decision problems. This motivates Aldous to introduce the stronger notion of synonymity based on prediction processes. Stronger still, Hoover--Keisler formalize what it means that $X^i$, $i=1,2$ have the same probabilistic properties. We establish that canonical representatives of the Hoover--Keisler equivalence classes are given precisely by the set of all Markov-martingale laws on a specific nested path space $\mathsf M_\infty$. As a consequence we obtain that, modulo Hoover--Keisler equivalence, the class of stochastic processes forms a Polish space. On this space, processes are topologically close iff they model similar probabilistic phenomena. In particular this means that their laws as well as the information encoded in the respective filtrations are similar. Importantly, compact sets of processes admit a Prohorov-type characterization. We also obtain that for every stochastic process, defined on some abstract basis, there exists a process with identical probabilistic properties which is defined on a standard Borel space.

math.PR↗