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Sensitivity of causal distributionally robust optimization

Yifan Jiang, Jan Obloj

TL;DR

This work derives the first-order sensitivity of the value of causal DRO with respect to the penalization parameter, i.e., the sensitivity to model uncertainty, and establishes a novel stochastic Fubini theorem.

Abstract

We study the causal distributionally robust optimization (DRO) in both discrete- and continuous- time settings. The framework captures model uncertainty, with potential models penalized in function of their adapted Wasserstein distance to a given reference model. Strength of the penalty is controlled using a real-valued parameter which, in the special case of an indicator penalty, is simply the radius of the uncertainty ball. Our main results derive the first-order sensitivity of the value of causal DRO with respect to the penalization parameter, i.e., we compute the sensitivity to model uncertainty. Moreover, we investigate the case where a martingale constraint is imposed on the underlying model, as is the case for pricing measures in mathematical finance. We introduce different scaling regimes, which allow us to obtain the continuous-time sensitivities as nontrivial limits of their discrete-time counterparts. We illustrate our results with examples. The sensitivities are naturally expressed using optional projections of Malliavin derivatives. To establish our results we obtain several novel results which are of independent interest. In particular, we introduce pathwise Malliavin derivatives and show these extend the classical notion. We also establish a novel stochastic Fubini theorem.

Sensitivity of causal distributionally robust optimization

TL;DR

This work derives the first-order sensitivity of the value of causal DRO with respect to the penalization parameter, i.e., the sensitivity to model uncertainty, and establishes a novel stochastic Fubini theorem.

Abstract

We study the causal distributionally robust optimization (DRO) in both discrete- and continuous- time settings. The framework captures model uncertainty, with potential models penalized in function of their adapted Wasserstein distance to a given reference model. Strength of the penalty is controlled using a real-valued parameter which, in the special case of an indicator penalty, is simply the radius of the uncertainty ball. Our main results derive the first-order sensitivity of the value of causal DRO with respect to the penalization parameter, i.e., we compute the sensitivity to model uncertainty. Moreover, we investigate the case where a martingale constraint is imposed on the underlying model, as is the case for pricing measures in mathematical finance. We introduce different scaling regimes, which allow us to obtain the continuous-time sensitivities as nontrivial limits of their discrete-time counterparts. We illustrate our results with examples. The sensitivities are naturally expressed using optional projections of Malliavin derivatives. To establish our results we obtain several novel results which are of independent interest. In particular, we introduce pathwise Malliavin derivatives and show these extend the classical notion. We also establish a novel stochastic Fubini theorem.
Paper Structure (18 sections, 15 theorems, 201 equations, 1 figure)

This paper contains 18 sections, 15 theorems, 201 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Causal optimal transport
  5. Some tools from stochastic calculus
  6. Classical Malliavin derivative
  7. Pathwise Malliavin derivative
  8. Forward integral against regular martingales
  9. Sensitivity of (bi)causal DRO
  10. Discrete-time results
  11. Continuous-time results
  12. Extensions and further results
  13. A general strategy of the proof
  14. Proofs of discrete-time results
  15. Proofs of continuous-time results
  16. ...and 3 more sections

Key Result

Proposition 2.2

Let $\mu,\nu\in \mathscr{M}(\mathcal{X})$. Then $\pi\in \Pi_{\mathfrak{bc}}(\mu,\nu)$ implies that $(X,Y)$ is an $(\mathcal{F}_{t}\otimes \mathcal{G}_{t})_{t\in I}$--martingale. Suppose additionally that $X$ and $Y$ have the martingale representation property under $\mu$ and $\nu$ respectively. Then

Figures (1)

  • Figure 4.1: Comparison of sensitivities of the Asian option \ref{['eq:Asianpayoff']} price under $\mu$, with $N=10$, as a function of strike $K$: parametric sensitivity with respect to the jump size $\mathfrak{j}$ and the non-parametric $\Upsilon_{\mathrm{Mart}}$ given in \ref{['eq:UpsilonMartAsian']}.

Theorems & Definitions (47)

  • Definition 2.1: acciaioCausalOptimalTransport2020
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 3.1: Clark--Ocone formula
  • Proposition 3.2
  • Definition 3.3: Pathwise Malliavin derivative
  • Definition 3.4: Left limit
  • Definition 3.5: Boundness preservation
  • Proposition 3.6
  • ...and 37 more