Law Invariant Coherent Risk Measures

• Law invariant coherent risk measures are convex functionals that depend solely on the distribution of outcomes and satisfy axioms like monotonicity, subadditivity, positive homogeneity, and translation equivariance.
• The Kusuoka representation expresses these measures as a supremum over integrated Average Value-at-Risk or spectral risk measures, yielding a unique minimal representation through stochastic order relations.
• This framework supports robust risk management practices in both atomless and atomic probability spaces, ensuring interpretable, computationally efficient models for financial and regulatory applications.

A law invariant coherent risk measure is a convex functional on a space of random variables that depends only on the probability law of the argument and satisfies the axioms of monotonicity, subadditivity (or convexity), positive homogeneity, and translation equivariance. These measures are central in risk management, quantifying the minimal capital required to mitigate uncertainty in financial, insurance, or operational settings, and provide robust, model-insensitive procedures as their evaluation is invariant under any probability-preserving transformation of outcomes. The formal analysis of such measures reveals a deep interplay between their representation, order-theoretic properties, duality, and regulatory significance, with the Kusuoka representation offering a canonical supremum structure over integrated Average Value-at-Risk (AV@R) or spectral risk measures.

1. Foundations and Formal Definition

A coherent risk measure $\rho: L_p \to \mathbb{R}$, where $L_p$ is a Lebesgue space, must satisfy:

• Monotonicity: $Z_1 \leq Z_2 \implies \rho(Z_1) \geq \rho(Z_2)$
• Subadditivity: $\rho(Z_1 + Z_2) \leq \rho(Z_1) + \rho(Z_2)$
• Positive homogeneity: $\rho(\lambda Z) = \lambda \rho(Z)$ for all $\lambda \geq 0$
• Translation invariance: $\rho(Z + m) = \rho(Z) - m$ for $m \in \mathbb{R}$

Law invariance requires that $\rho(Z_1) = \rho(Z_2)$ whenever $Z_1$ and $Z_2$ have the same distribution.

This property ensures that the risk measure responds only to the empirical distribution of outcomes, making it highly suited for model-agnostic and regulatory applications.

2. Kusuoka Representation and Minimality

The Kusuoka representation underpins the structure of law invariant coherent risk measures on atomless probability spaces. It states that any such $\rho$ admits the form

$$\rho(Z) = \sup_{\mu \in \mathcal{M}} \int_0^1 AV@R_\alpha(Z) \, d\mu(\alpha)$$

where $AV@R_\alpha$ is the Average Value-at-Risk at level $\alpha$, and $\mathcal{M}$ is a set of probability measures on $[0,1]$ with no mass at $1$.

An equivalent formulation uses spectral functions $\sigma$: $$\rho(Z) = \sup_{\sigma \in \Upsilon} \int_0^1 \sigma(u) F_Z^{-1}(u) du$$ where $F_Z^{-1}$ is the quantile function of $Z$, and $\sigma$ is $\mathbb{R}_+$-valued, monotone, and $\int_0^1 \sigma(u) du = 1$.

A key result (Pichler et al., 2012) establishes that every Kusuoka representation can be reduced to a minimal representation, where the set $\mathcal{M}_{\min}$ is the unique (in a weak* topology sense) smallest Kusuoka set such that no measure in $\mathcal{M}_{\min}$ can be omitted without changing $\rho$. Minimality is characterized through stochastic order relations: for measures $\mu_1, \mu_2$,

$$\mu_1 \preccurlyeq \mu_2 \iff \mu_1(\alpha) \geq \mu_2(\alpha)\ \text{for every } \alpha \in [0,1]$$

Non-dominated measures are extremal and constitute the essential building blocks of $\rho$'s Kusuoka representation.

3. Uniqueness, Duality, and Computational Aspects

Uniqueness of the minimal Kusuoka representation is established by mapping "exposed points" in the dual (weak*-compact) set of density functions via stochastic orders and explicit transforms between measures and spectral functions: $$\sigma(\tau) = \int_0^\tau (1 - \alpha)^{-1} d\mu(\alpha),\quad \mu(\alpha) = (1 - \alpha) \sigma(\alpha) + \int_0^\alpha \sigma(u) du$$ These transforms provide bijections between the representations in measure and function space.

From a computational perspective, the minimal Kusuoka representation prunes redundant elements via stochastic dominance and majorization, yielding a representation containing only those measures/spectra essential for optimizing $\rho(Z)$. This leads to sharp, non-redundant yet still interpretable decomposition of law invariant coherent risk measures—a crucial step for numerical stability and tractability (Pichler et al., 2012).

4. Extension to Probability Spaces with Atoms

While classical Kusuoka representations presume a nonatomic probability space, many practical problems deal with spaces containing atoms (e.g., finite scenario modeling, credit portfolios).

A risk measure $\hat{\rho}$ on such a space is regular if it restricts from a law invariant coherent measure on a standard atomless space. (Pichler et al., 2012) demonstrates that, under a technical condition (existence of appropriate monotone densities under measure-preserving maps), every law invariant coherent risk measure in this setting is regular and thus still enjoys a Kusuoka-type representation. However, if the atom structure obstructs such density construction, a Kusuoka representation may fail.

This result underlines the broad applicability of Kusuoka theory while precisely identifying its boundary: in most practical atomic models with equal probabilities, the Kusuoka representation survives intact, but there are also scenarios where representation fails due to atomic irregularities.

5. Practical Examples and Applications

Kusuoka minimal representation arises concretely in advanced financial risk measures:

Example class	Kusuoka/minimal representation structure	Parameters
Higher order risk measures	$$\rho(Z) = \sup\big\{\rho_\sigma(Z): \|\sigma\|_q = c\big\}$$	$1 < p < \infty$, $1/p + 1/q = 1$, $c > 1$
$p$-semideviation risk	$\rho(Z) = E[Z] + \lambda \|(Z-E[Z])_+\|_p$	$0 \leq \lambda \leq 1$, $p > 1$
Minimal Kusuoka by spectral norm	$$\rho(Z) = \sup_{\|\mu\|_q = c} \int_0^1 AV@R_\alpha(Z) d\mu(\alpha)$$

In both higher order and semideviation-based risk measures, the reduction to minimal Kusuoka representation leverages constraints on norms of spectra or measures, which are sharp, and this minimality is preserved regardless of the underlying space's atomic structure—provided the aforementioned regularity condition holds (Pichler et al., 2012).

6. Stochastic Orders and Order-theoretic Structure

Order relations underlie the uniqueness and efficiency of Kusuoka-type representations. First-order stochastic dominance for measures (and majorization for spectral functions) formalize eliminable redundancy:

• If measure $\mu_1$ is stochastically dominated by $\mu_2$, the spectral risk measure generated by $\mu_1$ is always weakly dominated by that generated by $\mu_2$, and $\mu_1$ can be removed from the Kusuoka set without affecting the supremum.
• The minimal Kusuoka set consists exactly of extremal, non-dominated measures.

This approach ensures not just uniqueness, but also the practical feasibility of constructing efficient representations usable in optimization, backtesting, or scenario analysis.

7. Significance and Theoretical Impact

The results in (Pichler et al., 2012) clarify that:

• Law invariant coherent risk measures admit Kusuoka representations in terms of mixtures (suprema) of AV@R or spectral risk measures.
• The minimal Kusuoka representation is unique and comprises the smallest possible set of extremal measures or spectra, identified via stochastic order.
• These advances extend to model spaces with atoms, encompassing a wide range of real-world applications.
• The methods enable risk managers and theoreticians to develop concise, interpretable, and computationally robust risk measurement schemes and to analyze their sensitivity and redundancy.

The Kusuoka framework and its minimal uniqueness property have become foundational tools in the mathematics of risk, optimization, and regulatory compliance.