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Quantifying Distributional Model Risk via Optimal Transport

Jose Blanchet, Karthyek R. A. Murthy

TL;DR

Quantifying Distributional Model Risk via Optimal Transport develops a general framework to bound expectations under model misspecification by constructing transport-based ambiguity sets around a baseline measure. It proves a strong duality result that reduces the infinite-dimensional worst-case problem to a univariate dual formulation, enabling tractable computation even for path-dependent functionals. The paper applies the theory to ruin probabilities, general first-passage problems, and ambiguity-averse decision making, including nonparametric calibration via Skorokhod embeddings, and discusses conditions for primal optimizer existence. The approach yields interpretable, tractable bounds and provides practical tools for risk management under model uncertainty, especially in stochastic-process contexts where KL-divergence is inapplicable.

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.

Quantifying Distributional Model Risk via Optimal Transport

TL;DR

Quantifying Distributional Model Risk via Optimal Transport develops a general framework to bound expectations under model misspecification by constructing transport-based ambiguity sets around a baseline measure. It proves a strong duality result that reduces the infinite-dimensional worst-case problem to a univariate dual formulation, enabling tractable computation even for path-dependent functionals. The paper applies the theory to ruin probabilities, general first-passage problems, and ambiguity-averse decision making, including nonparametric calibration via Skorokhod embeddings, and discusses conditions for primal optimizer existence. The approach yields interpretable, tractable bounds and provides practical tools for risk management under model uncertainty, especially in stochastic-process contexts where KL-divergence is inapplicable.

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.

Paper Structure

This paper contains 23 sections, 20 theorems, 204 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction.
  2. Our Main Result.
  3. Notation and Definitions.
  4. Primal Problem.
  5. The Dual Problem and Weak Duality.
  6. Main Result: Strong Duality Holds.
  7. Application of Theorem \ref{['THM-STRONG-DUALITY']} for computing worst-case probabilities.
  8. Proofs of Lemma \ref{['Lem-Optimality-lambda-Prob']} and Theorem \ref{['Thm-Prob-Reform']}.
  9. Computing ruin probabilities: A first example.
  10. Proof of the duality theorem.
  11. Duality in compact spaces.
  12. A note on additional notations and measurability.
  13. Extension of duality to non-compact spaces.
  14. On the existence of a primal optimal transport plan.
  15. A few more examples.
  16. ...and 8 more sections

Key Result

Theorem 1

Under the Assumptions (A1) and (A2), (a) $I=J$. In other words, (b) For any $\lambda\geq0,$ define $\phi_{_{\lambda}}:S\rightarrow \mathbb{R}\cup\{\infty\}$ as follows: There exists a dual optimizer of the form $(\lambda, \phi_{_{\lambda}}),$ for some $\lambda \geq 0.$ In addition, any feasible $\pi ^{\ast}\in\Phi_{\mu,\delta}$ and $(\lambda^{\ast},\phi_{_{\lambda^{\ast}}})\in\Lambda_{c,f}$ are

Figures (4)

  • Figure 1: Safety loading vs capital requirement for the Brownian model $R_B(t)$ (in red) and its robust counterpart (in blue) in Example \ref{['Eg-Ruin-prob-BM-approx']}. The objective is to keep the probability of ruin below 0.01.
  • Figure 2: Comparison of computation of ruin under baseline measure (in Fig(a)) and worst-case ruin (in Fig(b))
  • Figure 3: Capital requirement for various values of $\delta.$ The capital requirement is calculated to keep the worst-case probability of ruin under 0.01
  • Figure 4: A coupled path output by Algorithm 1

Theorems & Definitions (48)

  • Theorem 1
  • Remark 1: on the value of the dual problem
  • Remark 2: on the structure of primal optimal transport plan
  • Remark 3: On the uniqueness of a primal optimal transport plan
  • Remark 4: on $\varepsilon\xspace-$optimal transport plans
  • Remark 5
  • Lemma 2
  • Theorem 3
  • Remark 6
  • Lemma 4
  • ...and 38 more