G-Randomly Distorted Choquet Integrals

• G-Randomly Distorted Choquet Integrals are integrals modified by a stochastic distortion function measurable with respect to a sub–σ–algebra, enabling scenario-dependent risk evaluation.
• They generalize classical and deterministic distorted Choquet integrals by preserving key properties like monotonicity, comonotonic additivity, and translation invariance.
• This framework facilitates modeling conditional risk measures in finance and insurance by adapting to state-dependent information and uncertainties in model parameters.

A G-Randomly Distorted Choquet Integral (G-RDCI) is a Choquet integral wherein the distortion applied to a capacity is itself a (possibly random) function measurable with respect to a sub–σ–algebra G, allowing the distortion to vary stochastically across scenarios. G-RDCIs generalize both the classical Choquet integral and deterministic distorted Choquet integrals and serve as a mathematically robust tool for representing conditional risk measures in finance, insurance, and decision analysis under model and parameter uncertainty. They further admit canonical representations and satisfy crucial properties such as comonotonic additivity, monotonicity, and invariance under translation, thus enabling their use in both theoretical and applied risk modeling.

1. Formal Definition and Framework

Let $(\Omega, F)$ be a measurable space and %%%%1%%%% a capacity—i.e., a monotone set function with $c(\emptyset) = 0$, $c(\Omega) = 1$ and, typically, continuity from below. A random distortion is a mapping $\varphi^G: \Omega \times [0,1] \to [0,1]$ such that for each $\omega \in \Omega$, $t \mapsto \varphi^G(\omega, t)$ is non-decreasing, $\varphi^G(\omega, 0) = 0$, $\varphi^G(\omega, 1) = 1$, and for each $t$, $\omega \mapsto \varphi^G(\omega, t)$ is $G$-measurable with $G \subset F$ a sub-$\sigma$-algebra (encoding available information or regime).

For any bounded $F$-measurable random variable $X \in \chi(F)$, the G-randomly distorted Choquet integral is defined pointwise for each $\omega \in \Omega$ as

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

This construction recovers the classic Choquet integral for $\varphi^G(\omega, t) = t$ and the (deterministic) distorted Choquet integral when $\varphi^G(\omega, t)\equiv \varphi(t)$.

For step functions $X = \sum_{i=1}^n x_i 1_{A_i}$ with $x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$, and $A_i \in F$ disjoint, the explicit formula is

$$E_{(\varphi^G \circ c)}(X)(\omega) = \sum_{i=1}^n (x_i - x_{i+1}) \varphi^G(\omega, c(\cup_{k=1}^i A_k)),$$

with $x_{n+1} = 0$.

2. Core Properties

G-RDCIs satisfy the following axiomatic and structural properties (unless the distortion function or the capacity is allowed to be degenerate):

• Monotonicity: $X \leq Y$ pointwise implies $$E_{(\varphi^G \circ c)}(X)(\omega)\leq E_{(\varphi^G \circ c)}(Y)(\omega)$$ for all $\omega$.
• Translation (Cash) Invariance: For any constant $a\in\mathbb{R}$,

$$E_{(\varphi^G \circ c)}(a + X)(\omega) = a + E_{(\varphi^G \circ c)}(X)(\omega).$$

• Comonotonic Additivity: If $X$ and $Y$ are comonotonic (i.e., $$(X(\omega) - X(\omega')) (Y(\omega) - Y(\omega')) \geq 0$$ for all $(\omega,\omega')$), then

$$E_{(\varphi^G \circ c)}(X + Y) = E_{(\varphi^G \circ c)}(X) + E_{(\varphi^G \circ c)}(Y).$$

• Positive Homogeneity: The operator is positively homogeneous under suitable continuity and additivity conditions.
• Distribution Invariance: If $X$ and $Y$ have the same distribution with respect to $c$, their G-RDCIs coincide.

If $\varphi^G(\omega, \cdot)$ is concave for all $\omega$, the resulting risk measure is convex and monotone with respect to the stop-loss stochastic ordering. Under mild regularity, the G-RDCI is Lipschitz with respect to the uniform norm: $$\|E_{(\varphi^G \circ c)}(X) - E_{(\varphi^G \circ c)}(Y)\|_\infty \leq \|X-Y\|_\infty.$$

3. Representation of Conditional Risk Measures

The main representation theorem establishes that any G-conditional risk measure $\rho: \chi(F) \to \chi(G)$ that is

• monotone with respect to the $c$-first order stochastic dominance (i.e., $X \preceq_{st,c} Y$ implies $\rho(X) \leq \rho(Y)$),
• comonotonic additive,
• translation invariant, and
• positively homogeneous,

is necessarily a G-randomly distorted Choquet integral for a (unique) G-measurable random distortion function $\varphi^G$: $\rho(X) = E_{(\varphi^G \circ c)}(X).$ If the distortion is concave in $t$, this characterization tightens to monotonicity with respect to the stop-loss stochastic order, thus encompassing the entire family of convex law-invariant conditional risk measures on $(\Omega,F,c)$.

The construction of $\varphi^G$ proceeds by setting $\varphi^G(\omega, t) := \rho(1_A)(\omega)$ for any $A$ with $c(A) = t$, leveraging the property that risk measures are fully specified by values on indicators under monotonicity and comonotonic additivity.

4. Examples: Randomized VaR and AVaR

Randomized VaR: For $G = \{\emptyset, \Omega, A, A^c\}$ and $X = 1_B$, define

$$\varphi^G(\omega, t) := 1_{(1-\alpha, 1]}(t),\ \omega \in A;\quad \varphi^G(\omega, t) := 1_{(1-\beta, 1]}(t),\ \omega \in A^c,\quad (\alpha < \beta).$$

Then the G-RDCI is

$$E_{(\varphi^G \circ c)}(1_B)(\omega) = \begin{cases} \inf\{x: c(B) > 1-\alpha\}, & \omega \in A \ \inf\{x: c(B) > 1-\beta\}, & \omega \in A^c \end{cases}$$

recovering state-dependent VaR.

Randomized AVaR: For fixed levels $\alpha(\omega)$ measurable with respect to $G$, take

$$\varphi^G(\omega, t) = \frac{1}{1 - \alpha(\omega)} \min\{t, 1 - \alpha(\omega)\}.$$

The G-RDCI then yields a conditional AVaR, matching the robust average of quantiles over random endpoints.

Random Mixtures: By mixing deterministic and concave distortion functions according to $G$, the G-RDCI framework models risk measures that interpolate between VaR, AVaR, and other law-invariant functionals in a way responsive to scenario or expert-dependent information.

5. Mathematical Structure: Concave Random Distortions and Quantile Representation

Suppose $\varphi^G(\omega, \cdot)$ is concave for each $\omega$. Then, for any $X\in \chi(F)$ and $\omega\in\Omega$, the G-randomly distorted integral admits a quantile representation involving the (right-continuous) quantile function $r_X^{+}(t)$: $$E_{(\varphi^G \circ c)}(X)(\omega) = \varphi^G(\omega, 0^+) \cdot \sup_{t<1} r_X^{+}(t) + \int_0^1 \varphi^{G\prime}(\omega, 1-t) r_X^{+}(t) dt,$$ where $\varphi^{G\prime}$ denotes the right-derivative with respect to $t$.

The risk measure is then law-invariant (distribution-invariant under $c$) and convex.

6. Applications and Theoretical Implications

G-RDCIs provide a canonical form for all conditional risk measures satisfying law-invariance, comonotonic additivity, monotonicity, translation invariance, and positive homogeneity. They naturally extend classic regulatory and robust risk measures.

Application domains:

• Finance: State-dependent and ambiguity-sensitive forms of Value at Risk and Average Value at Risk are directly embedded within the G-RDCI framework, allowing for modeling risk aversion, scenario-dependent regulatory constraints, and expert opinion fusion.
• Insurance: Premium principles and loss functionals that incorporate heterogeneous, state-dependent risk aversion or regulatory constraints can be constructed by choosing appropriate G-measurable distortion functions.
• Aggregation of Opinions: In multi-expert systems or in Bayesian/frequentist blends, the G-RDCI enables the formal aggregation of scenario-specific risk assessments.

The flexibility of choosing $G$ adapts the model to dynamic information revelation or regime switching, as occurs in markets with regime-dependent behaviors, or in models where the distortion is influenced by stochastic volatility, environmental shocks, or abrupt changes in parameter regimes.

7. Comparison with Classical and Deterministic Distorted Integrals

Integral Type	Distortion Function	Measurability	Additivity
Choquet Integral	$\varphi(t) = t$	deterministic	Comonotonic additive
Distorted Choquet	$\varphi(t)$ deterministic	deterministic	Comonotonic additive
G-Randomly Distorted	$\varphi^G(\omega, t)$ $G$-measurable	random/profiled by $G$	Comonotonic additive

The G-RDCI therefore encompasses both classic and deterministically distorted integrals as special cases and extends representation results for law-invariant and comonotonic additive risk measures to the conditional, scenario-adaptive case.

Summary

G-Randomly Distorted Choquet Integrals generalize the theory of Choquet-type integrals by incorporating scenario-dependent, random distortion functions measurable with respect to a sub-$\sigma$-algebra G. They provide a unifying and axiomatic framework for representing conditional risk measures under model ambiguity, information flow, and statistical heterogeneity. The randomization of the distortion introduces a powerful toolset for constructing adaptive, robust risk functionals and paves the way for further extensions in both the mathematical theory of integration with non-additive measures and in practical risk aggregation under uncertainty.