Randomly Distorted Choquet Integrals

• Randomly distorted Choquet integrals are an extension of classical Choquet integrals that incorporate a stochastic distortion function for conditional, scenario-dependent aggregation.
• They preserve essential properties such as monotonicity, translation invariance, and comonotonic additivity, ensuring robust performance under uncertainty.
• This framework generalizes risk measures like Value-at-Risk and Average Value-at-Risk, accommodating heterogeneous information and enhancing uncertainty modeling.

Randomly distorted Choquet integrals generalize the classical Choquet integral by introducing a stochastic distortion function into the evaluation of non-additive set functions, with direct relevance to risk measurement, decision theory, and robust uncertainty modeling. The core idea is to replace the traditional (fixed) distortion function by a random variable measurable with respect to a prescribed sub-σ-algebra, thereby obtaining a conditional and scenario-dependent aggregation mechanism.

1. Mathematical Formulation

Let $(\Omega, \mathcal{F})$ be a measurable space, %%%%1%%%% a normalized, monotone capacity, and $\phi^G: \Omega \times [0,1] \to [0,1]$ a $G$-random distortion function, measurable with respect to a sub-σ-algebra $G \subset \mathcal{F}$ and satisfying:

• For each $\omega \in \Omega$, $t \mapsto \phi^G(\omega, t)$ is non-decreasing, $\phi^G(\omega, 0)=0$, $\phi^G(\omega,1)=1$.
• For each $t \in [0,1]$, $\omega \mapsto \phi^G(\omega, t)$ is $G$-measurable.

Given $X \in \chi(\mathcal{F})$, a bounded $\mathcal{F}$-measurable function, the randomly distorted Choquet integral is defined pointwise by:

$$E_{\phi^G \circ c}(X)(\omega) = \int_0^{+\infty} \phi^G(\omega, c(\{ X > x\})) dx + \int_{-\infty}^0 [\phi^G(\omega, c(\{ X > x\})) - 1] dx$$

This framework recovers the classical distorted (or non-random) Choquet integral as a special case when $\phi^G$ is deterministic and $c$ is a probability measure, but accommodates a wide range of randomization structures (e.g., partitions, mixtures, scenario-dependent distortions).

2. Fundamental Properties

The randomly distorted Choquet integral $E_{\phi^G \circ c}$ enjoys several robust properties under natural assumptions on $c$ and $\phi^G$:

• Distribution Invariance: If $c(\{X > x\}) = c(\{Y > x\})$ for all $x$, then $E_{\phi^G \circ c}(X) = E_{\phi^G \circ c}(Y)$.
• Monotonicity: For $X \leq Y$ pointwise, $E_{\phi^G \circ c}(X) \leq E_{\phi^G \circ c}(Y)$ for all $\omega$.
• Translation Invariance: For $a \in \mathbb{R}$, $$E_{\phi^G \circ c}(X+a) = a + E_{\phi^G \circ c}(X)$$.
• Positive Homogeneity: If $E_{\phi^G \circ c}$ is comonotonic-additive and Lipschitz, then $E_{\phi^G \circ c}(aX) = a E_{\phi^G \circ c}(X)$ for $a \geq 0$.
• Comonotonic Additivity: If $X$ and $Y$ are comonotonic, $$E_{\phi^G \circ c}(X + Y) = E_{\phi^G \circ c}(X) + E_{\phi^G \circ c}(Y)$$.

When the random distortion function $\phi^G(\omega, \cdot)$ is concave (in its second argument), the integral formalizes a risk-averse aggregation, and in this case, the corresponding risk measure is monotone with respect to stop-loss stochastic dominance (see technical definition in the paper).

3. Representation of Conditional Risk Measures

A pivotal result is the representation theorem for $G$-conditional risk measures. Suppose $\rho: \chi(\mathcal{F}) \to \chi(G)$ is a conditional risk measure satisfying:

1. Monotonicity w.r.t. first-order stochastic dominance (for $c$)
2. Comonotonic additivity
3. Translation invariance
4. Positive homogeneity

Then there exists a unique $G$-random distortion function $\phi^G$ such that, for all $X$:

$\rho(X) = E_{\phi^G \circ c}(X)$

If $\phi^G$ is concave, $\rho$ is monotone with respect to the stop-loss order.

This representation links abstract risk measurement directly to a random-distortion integral framework, facilitating analysis and computation under diverse informational regimes.

4. Illustrative Examples

(a) Randomized Value-at-Risk (VaR)

Let $G = \{ \varnothing, \Omega, A, A^c \}$, partitioning the space into two regions. Define:

$$\phi^G(\omega, t) = \begin{cases} 1_{(1-\alpha, 1]}(t) & \text{if } \omega \in A\ 1_{(1-\beta, 1]}(t) & \text{if } \omega \in A^c \end{cases} ~~ \text{with}~ \alpha < \beta$$

For $X=1_C$ (indicator variable), $E_{\phi^G \circ c}(X)(\omega)$ evaluates to different values depending on $c(C)$, $\alpha$, $\beta$, and whether $\omega \in A$ or $A^c$, thus “randomizing” the VaR threshold by $G$.

(b) Randomized Average Value-at-Risk (AVaR)

Define, for a partition $G$ with regions $A_1, ..., A_n$:

$$\phi^G(\omega, t) = \frac{1}{1-\alpha_i} \min \left\{ t, 1-\alpha_i \right\} \qquad \text{for}~ \omega \in A_i$$

This results in AVaR risk assessment that depends on the region/state $\omega$, capturing ambiguity in risk tolerances across heterogeneous information sets.

5. Connections and Implications

The randomly distorted Choquet framework unifies and extends prior work on Choquet integrals and risk measures:

• It generalizes classical risk functionals (VaR, AVaR) to settings where model ambiguity or expert disagreement is encoded via randomization of the distortion function.
• Comonotonic additivity and monotonicity persist under randomization, extending the robust aggregation properties of Choquet integrals to non-deterministic, conditional scenarios.
• The approach directly accommodates model uncertainty; the random distortion $\phi^G$ allows scenario-dependent aggregation, reflecting conditional information and expert opinions.
• For applications in finance, insurance, and robust decision making, randomly distorted Choquet integrals provide the analytic backbone for advanced risk assessment and aggregation under incomplete or diverse information.

6. Future Directions

Open directions include:

• Extension to dynamic, multi-period frameworks, where random distortion functions evolve with time or information.
• Deep integration with machine learning and expert aggregation, treating $\phi^G$ as an output of a learning process or as synthesized from multiple subjective assessments.
• Characterization and calibration of random distortion distributions to enhance empirical performance of conditional risk measures.
• Systematic paper of stochastic orderings and their interplay with conditional Choquet integrals, to ensure consistency and robustness in risk analysis.

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In summary, randomly distorted Choquet integrals $E_{\phi^G \circ c}$ create a powerful and flexible analytic framework for conditional and scenario-dependent aggregation in non-additive contexts, establishing rigorous foundations for conditional risk measurement and uncertainty quantification (Aldalbahi et al., 22 Sep 2025).