Quantifying Distributional Model Risk via Optimal Transport

Abstract

This paper deals with the problem of quantifying the impact of model misspecification when computing general expected values of interest. The methodology that we propose is applicable in great generality, in particular, we provide examples involving path-dependent expectations of stochastic processes. Our approach consists in computing bounds for the expectation of interest regardless of the probability measure used, as long as the measure lies within a prescribed tolerance measured in terms of a flexible class of distances from a suitable baseline model. These distances, based on optimal transportation between probability measures, include Wasserstein's distances as particular cases. The proposed methodology is well-suited for risk analysis, as we demonstrate with a number of applications. We also discuss how to estimate the tolerance region non-parametrically using Skorokhod-type embeddings in some of these applications.

Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning

“…is an extremal distribution that solves (6). For p > 1, the last constraint in (12) ensures that θ ij = 0 whenever α ij = 0 because otherwise α ij θ ij /α ij p evaluates to ∞. This implies that the set ν ∞ is empty.…”

Computational aspects of robust optimized certainty equivalents and option pricing

“…In [37] and [44], the authors focus on the case c(x, y) = |x − y| and µ 0 an empirical measure. The closest set of assumptions to ours is in [11]. Therein, the authors work on a general Polish space X, assume c to be lower semicontinuous and real-valued, and prove duality for (µ 0 -integrable) upper semicontinuous functions f .…”

“…We refer to Zhao and Guan [44] for a similar setting and another class of objective functions f . Blanchet and Murthy [11] and Gao and Kleywegt [27] obtain the result on a general Polish space and for lower semicontinuous cost functions c and upper semicontinuous integrable functions f . The proof given in the present paper is essentially more direct.…”

Asymptotic Analysis of a Multiclass Queueing Control Problem Under Heavy Traffic with Model Uncertainty

“…4. Recently, [14] quantified the impact of model misspecification when computing general expected values using an optimal transport cost instead of divergences. It would be interesting to tackle the QCP presented here and challenging the cµ rule using optimal transport tools.…”