Distortion Risk Metrics

Definition

A distortion (or g-entropic) risk metric is a coherent risk measure built from its dual “worst-case expectation over a risk envelope” representation: the risk equals the supremum of ${\mathbb{E}}_Q[X]$ over a convex, closed set $\mathcal{Q}$ of distributions absolutely continuous with respect to the nominal $P$, where the envelope is described by a convex function $g$ (an $f$-divergence / relative-entropy bound). CVaR, EVaR, and the total-variation-distance metric are all special cases obtained by choosing $g$; dixit2023risk exploits this shared structure so the optimizer is derived once and the metric swapped without re-derivation.

Key Equations

Dual (distributionally-robust) form of a coherent distortion metric:
$\rho (X)={\sup }_{Q\in \mathcal{Q}}{\mathbb{E}}_Q(X),\mathcal{Q}\subset \{Q\ll P\}\text{convex, closed (risk envelope)}.$
Via convex conjugacy the risk-constrained safety condition reformulates (Lemma 2) into a finite convex program in dual variables using the conjugate $g^{\ast }$ of the envelope describing function — gloss: each metric (CVaR: Radon–Nikodym bound $dQ/dP\le 1/(1-\alpha )$; EVaR: KL epigraph; TVD: total-variation ball) is just a different $g$.

Source Support

• dixit2023risk — the dual risk-envelope / $g$-entropic representation (Def. 2, Assumption 4), the convex-conjugate reformulation (Lemma 2), and the worked CVaR/EVaR/TVD envelopes for risk-averse obstacle-avoidance MPC.
• akella2024risk — coherence axioms and the representation theorem $\rho (X)={\sup }_{Q\in \mathcal{Q}}{\mathbb{E}}_Q(X)$ underpinning the distortion construction.

Related Topics

• coherent_risk_measures — distortion metrics are exactly the coherent measures via the envelope theorem.
• conditional_value_at_risk — the Radon–Nikodym-bounded envelope instance.
• entropic_value_at_risk — the KL-divergence (relative-entropy) envelope instance.
• chance_constraints — a distortion-risk bound gives a distribution-free convex inner approximation.

Open Questions

• Which envelope $g$ best balances conservatism and convex-program size when risk propagates through a nonlinear free-flying manipulator Jacobian rather than the linear plant of dixit2023risk?