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We consider multiperiod stochastic control problems with non-parametric uncertainty on the underlying probabilistic model. We derive a new metric on the space of probability measures, called the adapted $(p, \infty)$--Wasserstein distance $\mathcal{AW}_p^\infty$ with the following properties: (1) the adapted $(p, \infty)$--Wasserstein distance generates a topology that guarantees continuity of stochastic control problems and (2) the corresponding $\mathcal{AW}_p^\infty$-distributionally robust optimization (DRO) problem can be computed via a dynamic programming principle involving one-step Wasserstein-DRO problems. If the cost function is semi-separable, then we further show that a minimax theorem holds, even though balls with respect to $\mathcal{AW}_p^\infty$ are neither convex nor compact in general. We also derive first-order sensitivity results.

We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals $\mu$ and $\nu$. It has been empirically observed that the optimal coupling $\pi_\epsilon$ for the QOT problem has sparse support for small regularization parameter $\epsilon>0.$ In this article we provide the first quantitative description of this phenomenon in general dimension: we derive bounds on the size and on the location of the support of $\pi_\epsilon$ compared to the Monge coupling. Our analysis is based on pointwise bounds on the density of $\pi_\epsilon$ together with Minty's trick, which provides a quadratic detachment from the optimal transport duality gap. In the self-transport setting $\mu=\nu$ we obtain optimal rates of order $\epsilon^{\frac{1}{2+d}}.$

The Wasserstein distance $\mathcal{W}_p$ is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p$ extends this theory to laws of discrete time stochastic processes in their natural filtrations, making it particularly well suited for analyzing time-dependent stochastic optimization problems. While the topological differences between $\mathcal{A}\mathcal{W}_p$ and $\mathcal{W}_p$ are well understood, their differences as metrics remain largely unexplored beyond the trivial bound $\mathcal{W}_p\lesssim \mathcal{A}\mathcal{W}_p$. This paper closes this gap by providing upper bounds of $\mathcal{A}\mathcal{W}_p$ in terms of $\mathcal{W}_p$ through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of $\mathcal{W}_p$, Eder's modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on $\mathcal{W}_p$ automatically hold for $\mathcal{AW}_p$ under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality $\mathcal{A}\mathcal{W}_1\le C\sqrt{\mathcal{W}_1}$ on the set of measures that have Lipschitz kernels. Our work also reveals how smoothing of measures affects the adapted weak topology. In fact, we find that the topology induced by the smooth adapted Wasserstein distance exhibits a non-trivial interpolation property, which we characterize explicitly: it lies in between the adapted weak topology and the weak topology, and the inclusion is governed by the decay of the smoothing parameter.

Optimal transport and the Wasserstein distance $\mathcal{W}_p$ have recently seen a number of applications in the fields of statistics, machine learning, data science, and the physical sciences. These applications are however severely restricted by the curse of dimensionality, meaning that the number of data points needed to estimate these problems accurately increases exponentially in the dimension. To alleviate this problem, a number of variants of $\mathcal{W}_p$ have been introduced. We focus here on one of these variants, namely the max-sliced Wasserstein metric $\overline{\mathcal{W}}_p$. This metric reduces the high-dimensional minimization problem given by $\mathcal{W}_p$ to a maximum of one-dimensional measurements in an effort to overcome the curse of dimensionality. In this note we derive concentration results and upper bounds on the expectation of $\overline{\mathcal{W}}_p$ between the true and empirical measure on unbounded reproducing kernel Hilbert spaces. We show that, under quite generic assumptions, probability measures concentrate uniformly fast in one-dimensional subspaces, at (nearly) parametric rates. Our results rely on an improvement of currently known bounds for $\overline{\mathcal{W}}_p$ in the finite-dimensional case.

Given a collection of multidimensional pairs $\{(X_i,Y_i):1 \leq i\leq n\}$, we study the problem of projecting the associated suitably smoothed empirical measure onto the space of martingale couplings (i.e. distributions satisfying $\mathbb{E}[Y|X]=X$) using the adapted Wasserstein distance. We call the resulting distance the smoothed empirical martingale projection distance (SE-MPD), for which we obtain an explicit characterization. We also show that the space of martingale couplings remains invariant under the smoothing operation. We study the asymptotic limit of the SE-MPD, which converges at a parametric rate as the sample size increases if the pairs are either i.i.d. or satisfy appropriate mixing assumptions. Additional finite-sample results are also investigated. Using these results, we introduce a novel consistent martingale coupling hypothesis test, which we apply to test the existence of arbitrage opportunities in recently introduced neural network-based generative models for asset pricing calibration., Comment: 55 pages, 7 figures

We study a martingale Schr\"odinger bridge problem: given two probability distributions, find their martingale coupling with minimal relative entropy. Our main result provides Schr\"odinger potentials for this coupling. Namely, under certain conditions, the log-density of the optimal coupling is given by a triplet of real functions representing the marginal and martingale constraints. The potentials are also described as the solution of a dual problem.

We investigate an expected utility maximization problem under model uncertainty in a one-period financial market. We capture model uncertainty by replacing the baseline model $\mathbb{P}$ with an adverse choice from a Wasserstein ball of radius $k$ around $\mathbb{P}$ in the space of probability measures and consider the corresponding Wasserstein distributionally robust optimization problem. We show that optimal solutions converge to a strategy with minimal norm when uncertainty is increasingly large, i.e. when the radius $k$ tends to infinity., Comment: We have substantially generalized our main result, Theorem 1.1. We now consider general closed constraint sets $D$ and show that the optimal strategies converge to the one with minimal norm. In the case $D=\{w: \ge a\}$ for some $a>0$ we recover the re-weighted uniform strategy

Let $\mu$ be a probability measure on $\mathbb{R}^d$ and $\mu_N$ its empirical measure with sample size $N$. We prove a concentration inequality for the optimal transport cost between $\mu$ and $\mu_N$ for radial cost functions with polynomial local growth, that can have superpolynomial global growth. This result generalizes and improves upon estimates of Fournier and Guillin. The proof combines ideas from empirical process theory with known concentration rates for compactly supported $\mu$. By partitioning $\mathbb{R}^d$ into annuli, we infer a global estimate from local estimates on the annuli and conclude that the global estimate can be expressed as a sum of the local estimate and a mean-deviation probability for which efficient bounds are known., Comment: We added a section providing numerical evidence for our main result, and made some minor changes in the presentation

We study the classical problem of predicting an outcome variable, $Y$, using a linear combination of a $d$-dimensional covariate vector, $\mathbf{X}$. We are interested in linear predictors whose coefficients solve: % \begin{align*} \inf_{\boldsymbol{\beta} \in \mathbb{R}^d} \left( \mathbb{E}_{\mathbb{P}_n} \left[ \left(Y-\mathbf{X}^{\top}\beta \right)^r \right] \right)^{1/r} +\delta \, \rho\left(\boldsymbol{\beta}\right), \end{align*} where $\delta>0$ is a regularization parameter, $\rho:\mathbb{R}^d\to \mathbb{R}_+$ is a convex penalty function, $\mathbb{P}_n$ is the empirical distribution of the data, and $r\geq 1$. We present three sets of new results. First, we provide conditions under which linear predictors based on these estimators % solve a \emph{distributionally robust optimization} problem: they minimize the worst-case prediction error over distributions that are close to each other in a type of \emph{max-sliced Wasserstein metric}. Second, we provide a detailed finite-sample and asymptotic analysis of the statistical properties of the balls of distributions over which the worst-case prediction error is analyzed. Third, we use the distributionally robust optimality and our statistical analysis to present i) an oracle recommendation for the choice of regularization parameter, $\delta$, that guarantees good out-of-sample prediction error; and ii) a test-statistic to rank the out-of-sample performance of two different linear estimators. None of our results rely on sparsity assumptions about the true data generating process; thus, they broaden the scope of use of the square-root lasso and related estimators in prediction problems.

We analyze the effect of small changes in the underlying probabilistic model on the value of multi-period stochastic optimization problems and optimal stopping problems. We work in finite discrete time and measure these changes with the adapted Wasserstein distance. We prove explicit first-order approximations for both problems. Expected utility maximization is discussed as a special case., Comment: final version, accepted for publication in SIAM J. Financial Math

For probability measures $\mu,\nu$ and $\rho$ define the cost functionals \begin{align*} C(\mu,\rho):=\sup_{\pi\in \Pi(\mu,\rho)} \int \langle x,y\rangle\, \pi(dx,dy),\quad C(\nu,\rho):=\sup_{\pi\in \Pi(\nu,\rho)} \int \langle x,y\rangle\, \pi(dx,dy), \end{align*} where $\langle\cdot, \cdot\rangle$ denotes the scalar product and $\Pi(\cdot,\cdot)$ is the set of couplings. We show that two probability measures $\mu$ and $\nu$ on $\mathbb{R}^d$ with finite first moments are in convex order (i.e. $\mu\preceq_c\nu$) iff $C(\mu,\rho)\le C(\nu,\rho)$ holds for all probability measures $\rho$ on $\mathbb{R}^d$ with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of $\int f\,d\nu -\int f\,d\mu$ over all $1$-Lipschitz functions $f$, which is obtained through optimal transport duality and Brenier's theorem. Building on this result, we derive new proofs of well-known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.

We show that pointwise limits of semistatic trading strategies in discrete time are again semistatic strategies. The analysis is carried out in full generality for a two-period model, and under a probabilistic condition for multi-period, multi-stock models. Our result contrasts with a counterexample of Acciaio, Larsson and Schachermayer, and shows that their observation is due to a failure of integrability rather than instability of the semistatic form. Mathematically, our results relate to the decomposability of functions as studied in the context of Schr\"odinger bridges.

In a two-period financial market where a stock is traded dynamically and European options at maturity are traded statically, we study the so-called martingale Schr\"odinger bridge Q*; that is, the minimal-entropy martingale measure among all models calibrated to option prices. This minimization is shown to be in duality with an exponential utility maximization over semistatic portfolios. Under a technical condition on the physical measure P, we show that an optimal portfolio exists and provides an explicit solution for Q*. This result overcomes the remarkable issue of non-closedness of semistatic strategies discovered by Acciaio, Larsson and Schachermayer. Specifically, we exhibit a dense subset of calibrated martingale measures with particular properties to show that the portfolio in question has a well-defined and integrable option position.

We study the stability of entropically regularized optimal transport with respect to the marginals. Given marginals converging weakly, we establish a strong convergence for the Schr\"odinger potentials describing the density of the optimal couplings. When the marginals converge in total variation, the optimal couplings also converge in total variation. This is applied to show that Sinkhorn's algorithm converges in total variation when costs are quadratic and marginals are subgaussian, or more generally for all continuous costs satisfying an integrability condition.

We consider the optimal investment and marginal utility pricing problem of a risk averse agent and quantify their exposure to a small amount of model uncertainty. Specifically, we compute explicitly the first-order sensitivity of their value function, optimal investment policy and marginal option prices to model uncertainty. The latter is understood as replacing a baseline model $\mathbb{P}$ with an adverse choice from a small Wasserstein ball around $\mathbb{P}$ in the space of probability measures. Our sensitivities are thus fully non-parametric. We show that the results entangle the baseline model specification and the agent's risk attitudes. The sensitivities can behave in a non-monotone way as a function of the baseline model's Sharpe's ratio, the relative weighting of assets in an agent's portfolio can change and marginal prices can increase when an agent faces model uncertainty., Comment: Final version as published in "Mathematical Finance"

We study the potential functions that determine the optimal density for $\varepsilon$-entropically regularized optimal transport, the so-called Schr\"odinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. In the limit $\varepsilon\to0$ of vanishing regularization, strong compactness holds in $L^{1}$ and cluster points are Kantorovich potentials. In particular, the Schr\"odinger potentials converge in $L^{1}$ to the Kantorovich potentials as soon as the latter are unique. These results are proved for all continuous, integrable cost functions on Polish spaces. In the language of Schr\"odinger bridges, the limit corresponds to the small-noise regime., Comment: Forthcoming in 'Probability Theory and Related Fields'

Let $\pi\in \Pi(\mu,\nu)$ be a coupling between two probability measures $\mu$ and $\nu$ on a Polish space. In this article we propose and study a class of nonparametric measures of association between $\mu$ and $\nu$, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between $\nu$ and the disintegration $\pi_{x_1}$ of $\pi$ with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures $\mu$ and $\nu$. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglb\"ock, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces., Comment: Forthcoming in "Bernoulli"

Let $X$ be a compact metric space and let $v$ be a sub-additive capacity defined on $X$. We show that Lusin's theorem with respect to $v$ holds if and only if $v$ is continuous from above., Comment: Final version, accepted for publication in Proc. Am. Math. Soc

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least squares regression. We consider robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples., Comment: Forthcoming in "Proceedings of the Royal Society A"

A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g. Plug-Pichler - stochastic programming, Hellwig - game theory, Aldous - stability of optimal stopping, Hoover-Keisler - model theory). Remarkably, all these seemingly independent approaches define the same adapted weak topology in finite discrete time. Our first main result is to construct an adapted variant of the empirical measure that consistently estimates the laws of stochastic processes in full generality. A natural compatible metric for the weak adapted topology is the given by an adapted refinement of the Wasserstein distance, as established in the seminal works of Pflug-Pichler. Specifically, the adapted Wasserstein distance allows to control the error in stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. in a Lipschitz fashion. The second main result of this article yields quantitative bounds for the convergence of the adapted empirical measure with respect to adapted Wasserstein distance. Surprisingly, we obtain virtually the same optimal rates and concentration results that are known for the classical empirical measure wrt. Wasserstein distance.

Consider the empirical measure, $\hat{\mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $\mathbb{P}$ on the unit interval. For fixed $\mathbb{P}$ the Wasserstein distance between $\hat{\mathbb{P}}_N$ and $\mathbb{P}$ is a random variable on the sample space $[0,1]^N$. Our main result is that its normalised quantiles are asymptotically maximised when $\mathbb{P}$ is a convex combination between the uniform distribution supported on the two points $\{0,1\}$ and the uniform distribution on the unit interval $[0,1]$. This allows us to obtain explicit asymptotic confidence regions for the underlying measure $\mathbb{P}$. We also suggest extensions to higher dimensions with numerical evidence.

We show continuity of the martingale optimal transport optimisation problem as a functional of its marginals. This is achieved via an estimate on the projection in the nested/causal Wasserstein distance of an arbitrary coupling on to the set of martingale couplings with the same marginals. As a corollary we obtain an independent proof of sufficiency of the monotonicity principle established in [Beiglboeck, M., & Juillet, N. (2016). On a problem of optimal transport under marginal martingale constraints. Ann. Probab., 44 (2016), no. 1, 42106] for cost functions of polynomial growth.

In the frictionless discrete time financial market of Bouchard et al.(2015) we consider a trader who, due to regulatory requirements or internal risk management reasons, is required to hedge a claim $\xi$ in a risk-conservative way relative to a family of probability measures $\mathcal{P}$. We first describe the evolution of $\pi_t(\xi)$ - the superhedging price at time $t$ of the liability $\xi$ at maturity $T$ - via a dynamic programming principle and show that $\pi_t(\xi)$ can be seen as a concave envelope of $\pi_{t+1}(\xi)$ evaluated at today's prices. Then we consider an optimal investment problem for a trader who is rolling over her robust superhedge and phrase this as a robust maximisation problem, where the expected utility of inter-temporal consumption is optimised subject to a robust superhedging constraint. This utility maximisation is carrried out under a new family of measures $\mathcal{P}^u$, which no longer have to capture regulatory or institutional risk views but rather represent trader's subjective views on market dynamics. Under suitable assumptions on the trader's utility functions, we show that optimal investment and consumption strategies exist and further specify when, and in what sense, these may be unique.

We unify and establish equivalence between the pathwise and the quasi-sure approaches to robust modelling of financial markets in discrete time. In particular, we prove a Fundamental Theorem of Asset Pricing and a Superhedging Theorem, which encompass the formulations of [Bouchard, B., & Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. The Annals of Applied Probability, 25(2), 823-859] and [Burzoni, M., Frittelli, M., Hou, Z., Maggis, M., & Obloj, J. (2019). Pointwise arbitrage pricing theory in discrete time. Mathematics of Operations Research]. In bringing the two streams of literature together, we also examine and relate their many different notions of arbitrage. We also clarify the relation between robust and classical $\mathbb{P}$-specific results. Furthermore, we prove when a superhedging property w.r.t. the set of martingale measures supported on a set of paths $\Omega$ may be extended to a pathwise superhedging on $\Omega$ without changing the superhedging price.

We consider statistical estimation of superhedging prices using historical stock returns in a frictionless market with d traded assets. We introduce a plugin estimator based on empirical measures and show it is consistent but lacks suitable robustness. To address this we propose novel estimators which use a larger set of martingale measures defined through a tradeoff between the radius of Wasserstein balls around the empirical measure and the allowed norm of martingale densities. We establish consistency and robustness of these estimators and argue that they offer a superior performance relative to the plugin estimator. We generalise the results by replacing the superhedging criterion with acceptance relative to a risk measure. We further extend our study, in part, to the case of markets with traded options, to a multiperiod setting and to settings with model uncertainty. We also study convergence rates of estimators and convergence of superhedging strategies., Comment: This work will appear in the Annals of Statistics. The above version merges the main paper to appear in print and its online supplement