Coherent Risk Measures

Definition

A risk measure is a functional $\rho :\mathfrak{F}\to \mathbb{R}$ that maps a (cost) random variable to a single real number summarizing how undesirable its distribution is. It is coherent (Artzner et al. 1999) when it satisfies four axioms — monotonicity, translation invariance, positive homogeneity, and subadditivity — which together encode the “rational” risk-ordering properties a sensible safety assessment should obey; positive homogeneity and subadditivity jointly imply convexity. Every coherent risk measure admits a dual (worst-case-expectation) representation over a convex closed set of probability measures (the risk envelope), so coherence is equivalent to a form of distributional robustness. This is a general decision-theoretic construct (finance / operations-research origin); it is regime-agnostic and carries no space-dynamics assumption — neither source models a free-flying or free-floating base, so adopting it here is a modelling choice, not an inherited result.

Key Equations

Symbols per notation.md.

Note: $\rho$, the cost random variable $X$ (a.k.a. $Z$), the confidence level $\alpha$ (listed in notation.md as the CVaR level), and the risk envelope $\mathcal{Q}$ are risk-layer symbols not all tabulated centrally; they follow the cited sources. The conditional-tail special case ${\mathrm{CVaR}}_{\alpha }$ lives on conditional_value_at_risk.

The four coherence axioms, for cost variables $X,X^{\prime }$ and scalar $c\in \mathbb{R}$, $\lambda \ge 0$ (Artzner et al. 1999; dixit2023risk Def. 1):

$$\begin{array}{rl}\text{(monotonicity)}& X\le X^{\prime }\Rightarrow \rho (X)\le \rho (X^{\prime }),\\ \text{(translation invariance)}& \rho (X+c)=\rho (X)+c,\\ \text{(positive homogeneity)}& \rho (\lambda X)=\lambda \rho (X),\\ \text{(subadditivity)}& \rho (X+X^{\prime })\le \rho (X)+\rho (X^{\prime }).\end{array}$$

Dual / representation theorem — every coherent risk measure is a worst-case expectation over a convex, closed risk envelope $\mathcal{Q}\subset \{Q\ll P\}$ (dixit2023risk Def. 2; majumdar2017how writes the finite-support form $\rho (Z)={\max }_{p\in \mathcal{P}}{\mathbb{E}}_p[Z]$):

$$\rho (X)=\underset{Q\in \mathcal{Q}}{\sup }{\mathbb{E}}_Q[X].$$

Source Support

• dixit2023risk — gives the compact formal Definition 1 (the four axioms) and Definition 2 (the dual risk-envelope representation), then builds a risk-averse MPC for dynamic obstacle avoidance admitting any coherent measure (CVaR, EVaR, total-variation / $g$-entropic). Application examples are a planar/aerial vehicle (drone) under process + measurement noise — terrestrial/aerial, not a space base.
• majumdar2017how — merged from coherent_risk_metrics. The axiomatic-framework paper: argues robot risk metrics should satisfy axioms A1–A6, identifies A1–A4 as exactly the coherent class, and states the universal representation theorem with the risk envelope $\mathcal{P}$. Examples are autonomous cars (automotive/ground), not space robots.

Related Topics

• coherent_risk_metrics — the same notion under the “metrics” naming (majumdar2017how’s term); this page is the merge target and they should be kept synonymous.
• conditional_value_at_risk — CVaR is the canonical coherent measure; every coherent (distortion) measure is a CVaR mixture, so CVaR is the building block of the class.
• chance_constraints — chance constraints correspond to thresholding the non-coherent VaR (${\mathrm{VaR}}_{\alpha }(X)\le 0\iff \mathbb{P}[X>0]\le \alpha$); coherent measures are the convex, subadditive alternative.
• risk_aware_mpc — dixit2023risk plugs a coherent measure into the MPC stage/terminal constraints, exploiting the dual form to keep the program convex.
• time_consistency — a static coherent measure is not automatically time-consistent over a horizon; multistage use requires nested/compounded one-step measures (majumdar2017how §5).

Open Questions

• Both sources are regime-agnostic and demonstrated on ground/aerial vehicles; does the convex dual representation buy anything specific for our free-flying manipulator, where the dominant “cost” $X$ is singularity proximity (${\sigma }_G={\sigma }_{\min }(\mathbf{\Gamma })$) rather than obstacle distance?
• dixit2023risk preserves convexity of the MPC only because its dynamics are linear; the FFSM circumcentroidal dynamics are nonlinear — does the risk-envelope reformulation remain tractable, or only after linearization?
• majumdar2017how notes subadditivity (diversification) is “unclear” for low-level control tasks; is base-attitude vs. EE-pose risk genuinely diversifiable for a coordinated free-flying controller, or are these comonotone (rise/fall together)?