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Scaling limits of multi-period distributionally robust optimization problems

Max Nendel, Ariel Neufeld, Kyunghyun Park, Alessandro Sgarabottolo

TL;DR

The paper studies the scaling limit of multi-period distributionally robust optimization under Wasserstein uncertainty, proving that as the period length shrinks the resulting value evolves as a strongly continuous monotone semigroup $ olinebreak ( olinebreak ext{S}(t))_{t olinebreak o olinebreak 0}$ on ${ m C}_{ m b}$ with infinitesimal generator $ olinebreak ext{L}f= olinebreak ext{inf}_{a olinebreak A} olinebreak ext{L}^a f+mig\| abla figig Vert$. The generator decomposes into a non-robust part $ ext{inf}_a ext{L}^a f$ plus a Wasserstein-induced perturbation $mig\| abla figig Vert$, linking DRO to nonlinear PDEs. The authors show that the semigroup solutions are viscosity solutions to nonlinear HJB-Isaacs-type equations and connect these to g-expectations and drift-uncertainty robust optimization, including continuous-time robust control formulations for Itô processes. This yields a principled bridge from nonparametric Wasserstein DRO to parametric continuous-time robust optimization, enabling PDE-based analysis and computation of robust strategies. The work also establishes a detailed proof framework via dyadic partition limits and a best-case auxiliary operator, clarifying how dynamic consistency emerges in the scaling limit.

Abstract

We examine the scaling limit of multi-period distributionally robust optimization (DRO) problems via a semigroup approach. Each period involves a worst-case maximization over distributions in a Wasserstein ball around the transition probability of a reference process with radius proportional to the length of the period, and the multi-period DRO problem arises through its sequential composition. We show that the scaling limit of the multi-period DRO, as the length of each period tends to zero, is a strongly continuous monotone semigroup on $\mathrm{C_b}$. Furthermore, we show that its infinitesimal generator is equal to the generator associated with the non-robust scaling limit plus an additional perturbation term induced by the Wasserstein uncertainty. As an application, we show that when the reference process follows an Itô process, the viscosity solution of the associated nonlinear PDE coincides with the value of continuous-time robust optimization problems under parametric uncertainty.

Scaling limits of multi-period distributionally robust optimization problems

TL;DR

The paper studies the scaling limit of multi-period distributionally robust optimization under Wasserstein uncertainty, proving that as the period length shrinks the resulting value evolves as a strongly continuous monotone semigroup on with infinitesimal generator . The generator decomposes into a non-robust part plus a Wasserstein-induced perturbation , linking DRO to nonlinear PDEs. The authors show that the semigroup solutions are viscosity solutions to nonlinear HJB-Isaacs-type equations and connect these to g-expectations and drift-uncertainty robust optimization, including continuous-time robust control formulations for Itô processes. This yields a principled bridge from nonparametric Wasserstein DRO to parametric continuous-time robust optimization, enabling PDE-based analysis and computation of robust strategies. The work also establishes a detailed proof framework via dyadic partition limits and a best-case auxiliary operator, clarifying how dynamic consistency emerges in the scaling limit.

Abstract

We examine the scaling limit of multi-period distributionally robust optimization (DRO) problems via a semigroup approach. Each period involves a worst-case maximization over distributions in a Wasserstein ball around the transition probability of a reference process with radius proportional to the length of the period, and the multi-period DRO problem arises through its sequential composition. We show that the scaling limit of the multi-period DRO, as the length of each period tends to zero, is a strongly continuous monotone semigroup on . Furthermore, we show that its infinitesimal generator is equal to the generator associated with the non-robust scaling limit plus an additional perturbation term induced by the Wasserstein uncertainty. As an application, we show that when the reference process follows an Itô process, the viscosity solution of the associated nonlinear PDE coincides with the value of continuous-time robust optimization problems under parametric uncertainty.

Paper Structure

This paper contains 18 sections, 15 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Construction of the scaling limit of multi-period DROs
  3. Notation and preliminaries
  4. The controlled reference Markov process and its transition semigroup
  5. Single-period distributionally robust optimization
  6. Multi-period DRO and its scaling limit
  7. Connection to nonlinear expectations and robust optimization under drift uncertainty.
  8. Nonlinear PDEs arising from the semigroup $\mathcal{S}$ and their viscosity solutions
  9. Connection to $g$-expectation
  10. Connection to robust optimization under drift uncertainty
  11. Best-case maximization
  12. Proofs of results in Section \ref{['sec:DRO']}
  13. Proofs of results in Section \ref{['sec:multi.DRO']}
  14. Proofs of results in Section \ref{['sec:apply']}
  15. Supplementary statements
  16. ...and 3 more sections

Key Result

Proposition 2.4

Suppose that Assumption ass:general holds. In particular, choosing $m=0$, these properties also hold for the family $T$ instead of $I$.

Figures (1)

  • Figure 1: Multi-period DRO problem as iteration of the one-step DRO operator $I$.

Theorems & Definitions (42)

  • Example 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Remark 2.9
  • Theorem 2.10
  • Definition 3.1
  • Definition 3.2
  • ...and 32 more