Systemic Risk Measures with Market Volatility

Abstract

Systemic risk measures are crucial for the stability of financial markets. However, traditional frameworks, such as those relying on fixed exponent spaces, rest on an assumption of globally consistent moment conditions, which fails to capture the time-varying and state-dependent nature of financial market volatility. In this paper, we propose a new framework for systemic risk measurement on the variable-exponent Bochner–Lebesgue space $L^{p(·)}$, where the exponent $p(·)$ is a random variable rather than a deterministic constant parameter. The variable exponent $p(·)$ can be regarded as an endogenous risk-sensitivity adjuster. This design imbues the systemic risk measure with inherent adaptability, enabling it to move beyond applying a fixed scale across all market states and instead allowing its criterion to adjust dynamically in response to the evolving nature of risk amid shifts in market conditions. By constructing suitable deterministic auxiliary functions and single-firm risk measures, we decompose the quantification of systemic risk in $L^{p(·)}$ into two sequential steps, ultimately deriving its dual representations. Several examples are provided to illustrate the theoretical results.

1. Introduction

Systemic risk refers to the risk arising from macroeconomic shocks, policy changes, or similar aggregate disturbances whose effects spread throughout the market or financial system. Due to its contagious and destructive nature, systemic risk can undermine market confidence, cause sharp declines in asset prices, and ultimately trigger financial crises. As a result, systemic risk represents the greatest threat to financial stability, making its accurate measurement essential for protecting economic security and maintaining public trust. Systemic risk measurement provides a precise assessment of financial system vulnerability under extreme conditions and offers regulators a quantitative basis for setting capital buffers and liquidity requirements. Additionally, such measurement enables financial institutions to optimize asset allocation and prevent excessive risk accumulation. Furthermore, it helps identify critical nodes within the risk network, thereby facilitating targeted supervision. Central to this effort is the development of a robust framework for quantifying systemic risk.

Systemic risk measures were initially formalized through an axiomatic approach by Chen et al. [1], who demonstrated that while offsetting losses on individual assets with gains elsewhere constitutes rational portfolio management, this practice is inappropriate for macroeconomic assessment. This is because heterogeneous equity ownership structures across financial institutions render the aggregate social impact non-additive. Rejecting the reductionist view of the financial system as a single composite portfolio, Chen et al. [1] argued that monetary risk measures cannot be mechanically applied to economy-wide aggregates without accounting for inter-institutional interactions. Consequently, they decomposed systemic risk into an aggregation functional and a scalar risk component, establishing a mechanism to attribute individual institutions’ contributions. Subsequent developments include Kromer et al. [2], who extended this axiomatic framework to general probability spaces. Complementing this line of research, Doldi and Frittelli [3], Kromer et al. [4], and Hoffmann et al. [5] examined dynamic systemic risk measures and their associated time consistency properties from multiple theoretical perspectives. Additionally, Ararat and Rudloff [6] derived dual representations for these measures, while Gong and Hu [7] investigated dynamic risk measures under uncertainty. Additional contributions to systemic risk measurement include the works of Acharya et al. [8], Armenti et al. [9], Biagini et al. [10], Brunnermeier and Cheridito [11], Farkas and Lucescu [12], Feinstein et al. [13], Gauthier et al. [14], Tarashev et al. [15], and the references therein.

Recent seismic shifts in the global economic order, including geopolitical tensions and the ongoing retreat from globalization, have significantly increased price volatility. However, existing systemic risk measures remain fundamentally ill-suited to volatile market conditions. Current quantification methodologies have three critical limitations. First, conventional approaches rely predominantly on historical data. Yet, markets now exhibit low-frequency, high-severity loss dynamics, causing regulators to systematically underestimate risk exposures during tail events. Second, parameter estimation lags significantly, preventing the timely capture of market sentiment. Third, elevated high-frequency market volatility distorts fundamental asset pricing mechanisms, triggers liquidity crises, and simultaneously amplifies measurement errors—ultimately causing risk assessments to diverge significantly from realized losses. Therefore, developing a systemic risk framework specifically designed for volatile market environments has become essential to safeguarding financial stability. Beyond aggregate macroeconomic shocks, the time-varying nature of market volatility may itself be rooted in market-microstructure frictions and strategic interactions among informed agents. Recent work by Daher and Damrah [16] demonstrates that strategic competition between insiders with public information and overconfident market makers generates endogenous, state-dependent volatility dynamics. This provides a game-theoretic micro-foundation for the volatility indicator V introduced in Definition 1, suggesting that the randomness of the variable exponent $p(·)=g\left(V\right(·\left)\right)$ can be linked to behavioral and informational drivers rather than purely exogenous statistical specifications.

In this paper, we investigate systemic risk measures on the variable-exponent Bochner–Lebesgue space $L^{p(·)}$. The framework of this study is based on the following mapping:

$$\begin{array}{c}\hfill \rho :L^{p(·)}\ni f\mapsto \rho \left(f\right)\in \mathbb{R}\cup \{+\infty \}\end{array}$$

which quantifies the characteristics of systemic risk in the presence of market volatility. For research on volatility in risk measures, see Mitra [17] and Cheung and Yuen [18]. A core characteristic of financial market volatility is its time-varying and state-dependent nature. During tranquil periods, the distribution of asset returns is relatively concentrated, with risks primarily manifested as variations around the mean. In contrast, during periods of stress, the market experiences not only intensified fluctuations but, more critically, a shift in the shape of its distribution. In particular, the pronounced emergence of tail risk substantially increases the probability of extreme losses. The classical risk measurement framework, which relies on the fixed-exponent space, implicitly assumes a globally uniform moment condition. It presumes that the metric for the magnitude of risk—that is, the emphasis placed on the moments of loss—remains constant regardless of the market state. This assumption clearly fails to capture the dynamic features described above. A space with a fixed q either overemphasizes the tail during calm periods (if q is relatively large) or underestimates extreme losses during crises (if q is relatively small). The study of systemic risk measures on the space $L^{p(·)}$ is precisely aimed at incorporating the quality—not just the quantity—of volatility into risk measurement. Here, the variable exponent $p(·)$ is formally modeled as an endogenous risk-sensitivity adjuster driven by a market volatility indicator $V\left(\omega \right)$ via a measurable mapping g, i.e., $p\left(\omega \right)=g\left(V\right(\omega \left)\right)$. The monotonicity of g ensures that when market volatility rises and is accompanied by an accumulation of tail risk, $p(·)$ increases accordingly, causing the norm of the space to penalize large losses more severely, thereby automatically enhancing the sensitivity of the risk measure to extreme events. Conversely, during periods of lower volatility and milder return distributions, $p(·)$ decreases, shifting the focus of the measure toward the central tendency of losses. The precise mathematical specification of this volatility-driven mechanism is provided in Section 2.

It should be noted that the scalar variable-exponent Lebesgue space $L^{p(·)}$ was first introduced by Orlicz [19]. Subsequently, Cheng and Xu [20] introduced and systematically studied the Banach-space-valued Bochner–Lebesgue space with variable exponent, denoted by $L^{p(·)}(\Omega ,E)$, and for the first time provided a characterization of its dual space. Further studies on $L^{p(·)}$ include those by Almeida et al. [21], Diening et al. [22], Harjulehto et al. [23], Hästö [24], Kempka [25,26], Kováčik and Rákosník [27], Xu [28,29], and the references therein.

It turns out that constructing systemic risk measures on the space $L^{p(·)}$ can be decomposed into two sequential steps. First, a deterministic mapping dynamically normalizes systemic risk positions, reducing multi-factor systemic risk to a univariate measurable quantity driven by a single factor, thereby standardizing the impact of market fluctuations. Second, a single-firm risk measure is introduced to evaluate normalized positions. By decoupling the systemic risk structure in a staged manner, this approach retains the conventional ability to capture common market risk. Additionally, its modular design enhances cross-sector comparability and allows for dynamic recalibration, thereby improving the precision of systemic risk measurement in volatile markets.

The principal contribution of this paper is threefold. First, we introduce the variable-exponent Bochner–Lebesgue space $L^{p(·)}$—where the random exponent $p(·)$ encodes dynamic market volatility—as the mathematical domain for systemic risk positions. This appears to be the first application of such function spaces to systemic risk analysis. Second, within this framework, we develop systemic risk measures that inherently incorporate market volatility dynamics. We further prove a structural decomposition theorem: any systemic risk measure in $L^{p(·)}$ can be decomposed into a convex deterministic component and a single-firm risk component, providing novel insights into risk composition under volatile market conditions. Third, we derive the dual representations of these models by applying duality theory for $L^{p(·)}$. Obtaining these dual representations presents particular methodological challenges, as significant theoretical gaps exist between the established characterization theory of dual spaces with variable exponents and our derived representations. To bridge these gaps, we employ a novel approach involving carefully designed construction procedures.

The remainder of this paper is organized as follows. In Section 2, we briefly review the definition and main properties of the variable-exponent Bochner–Lebesgue space $L^{p(·)}$. In Section 3, we develop the definitions of systemic risk measures in $L^{p(·)}$ as well as the definitions of the convex deterministic function and single-firm risk measures. Section 4 discusses the construction of the systemic risk measures in $L^{p(·)}$. Section 5 is devoted to the dual representations of systemic risk measures in $L^{p(·)}$. Finally, in Section 6, examples of systemic risk measures in $L^{p(·)}$ are presented.

2. Preliminary Information

In this section, we briefly introduce the definition and main properties of variable-exponent Bochner–Lebesgue spaces, along with some preliminary information that will be used throughout this paper.

Let $(\Omega ,\mathcal{F},\mu )$ be a complete probability space. Let E be a given reflexive Banach space with norm $\parallel ·\parallel$ and dual space $E^{*}$ having the Radon–Nikodým property. We assume that $E^{*}$ is partially ordered by a given cone $K_0$ and E is partially ordered by K, where $K:=\{x\in E|\langle y,x\rangle \ge 0\mathrm{for}\mathrm{any}y\in K_0\}$ is the positive dual cone of $K_0$. We also suppose that the numéraire asset is some interior point $z\in int\left(K\right)$.

Remark 1.

The partial order relation ${\ge }_K$ is defined as follows, for any $x,y\in E$:

$$x{\ge }_{K}y\iff x-y\in K.$$

Remark 2.

The cone K consists of the ‘admissible’ loss functionals. When managing a diversified portfolio, investors do not require each individual investment to be profitable; some investments may incur losses, provided that the overall loss position remains acceptable. In the context of systemic risk, K is interpreted as a solvency set of loss positions: a position $x\in K$ means that the loss is acceptable (or ‘covered’) from the regulator’s perspective. Thus, larger positions in the partial order ${\ge }_K$ correspond to worse loss scenarios.

Definition 1

(Market volatility indicator). Let $V:\Omega \to [0,\infty )$ be an $\mathcal{F}$-measurable random variable representing a market volatility indicator (e.g., the implied volatility index such as VIX, realized volatility, or a systemic stress index constructed from cross-sectional asset returns).

Banach-space-valued Bochner–Lebesgue spaces with variable exponents were first introduced by Cheng and Xu [20]. We now recall the definition and related properties of these spaces. We denote the set of all $\mathcal{F}$-measurable functions $p(·):\Omega \to [1,\infty )$ by $\mathcal{S}(\Omega ,\mu )$; these functions are called variable-exponent functions on $\Omega$. For a function $p(·)\in \mathcal{S}(\Omega ,\mu )$, we define $p^{\prime }(·)\in \mathcal{S}(\Omega ,\mu )$ by $1/p\left(\omega \right)+1/p^{\prime }\left(\omega \right)=1$. The following definitions and properties are taken from the work of Cheng and Xu [20].

Definition 2.

A function $f:\Omega \to E$ is strongly $\mathcal{F}$-measurable if there exists a sequence ${\left\{f_n\right\}}_{n\ge 1}$ of μ-simple functions converging to f μ-almost everywhere.

Assumption 1

(Volatility-driven exponent). The variable exponent $p:\Omega \to [1,\infty )$ is generated by the market volatility indicator V introduced in Definition 1 via a measurable mapping $g:[0,\infty )\to [1,\infty )$, i.e.,

$$p\left(\omega \right)=g\left(V\left(\omega \right)\right),\omega \in \Omega .$$

(1)

We assume that g satisfies the following properties:

(G1) 

Monotonicity: g is non-decreasing. This ensures that higher market volatility leads to a larger exponent $p(·)$, intensifying the penalty on tail losses.

(G2) 

Regularity: g is Borel measurable and satisfies $g\left(v\right)\in [1,\infty )$ for all $v\in [0,\infty )$.

(G3) 

Boundedness from below: ${inf}_{v\ge 0}g\left(v\right)=p_{min}\ge 1$, ensuring that $L^{p(·)}$ remains a Banach space.

Definition 3.

The variable-exponent Bochner–Lebesgue space, denoted by $L^{p(·)}:=L^{p(·)}(\Omega ,E)$, is the collection of all strongly $\mathcal{F}$-measurable functions $f:\Omega \to E$ endowed with the norm

$${\parallel f\parallel }_{L^{p(·)}}:=inf\{\lambda >0|{\rho }_{p(·)}(f/\lambda )\le 1\}$$

where

$${\rho }_{p(·)}\left(f\right):={\mathit{\int }}_{\Omega }{\parallel f\left(\omega \right)\parallel }^{p\left(\omega \right)}d\mu \left(\omega \right)\mathrm{and}p(·)\in \mathcal{S}(\Omega ,\mu ).$$

In what follows, we always assume that $p(·)$ satisfies Assumption 1, so that its randomness is driven by the market volatility indicator V via the mapping g. The partial order $f{\ge }_{K}g$ on variable-exponent Bochner–Lebesgue space is defined as $f\left(\omega \right){\ge }_{K}g\left(\omega \right)$ for any $\omega \in \Omega$. Throughout this paper, all order relations and pointwise identities involving elements of $L^{p(·)}$ are understood to hold $\mu$-almost everywhere.

Assumption 2.

We assume that the variable exponent $p(·)$ satisfies

$$1<p^{-}\le p^{+}<\infty ,\mathit{where}{p}^{-}:=\underset{\omega \in \Omega }{essinf}p\left(\omega \right),p^{+}:=\underset{\omega \in \Omega }{esssup}p\left(\omega \right).$$

(2)

Under this assumption, the space $L^{p(·)}(\Omega ,E)$ is reflexive, and its dual is linearly isometric to $L^{p^{\prime }(·)}(\Omega ,E^{*})$ via the map $g\mapsto V_g$ in Remark 4.

Remark 3.

The functional form of g encodes the regulator’s or market participant’s risk attitude toward volatility. For instance:

• A capped linear specification,

  $$g\left(v\right)=min\{p_0+\kappa v,p_1\},1\le p_0<p_1<\infty ,\kappa >0,$$
  implies a proportional response of risk sensitivity to volatility up to the saturation level $p_1$; the cap guarantees $p^{+}\le p_1<\infty$, in line with Assumption 2. Equivalently, one may retain the purely linear form $g\left(v\right)=p_0+\kappa v$ under the additional assumption that the volatility indicator is essentially bounded, $V\le v_{max}$ a.s., which yields $p^{+}\le p_0+\kappa v_{max}<\infty$.
• A threshold specification

  $$g\left(v\right)=\left\{\begin{array}{cc}{p}_0,\hfill & v\le v^{*},\hfill \\ p_1,\hfill & v>v^{*},\hfill \end{array}\right.$$

  (3)
  with $1\le p_0<p_1<\infty$, captures a regime-switching risk appetite where the risk measure abruptly tightens during stress periods (e.g., $v^{*}$ corresponds to a systemic stress threshold).

Under Assumption 1, the randomness of $p(·)$ is endogenously inherited from the stochastic structure of V. Thus, the space $L^{p(·)}$ adapts its local integrability to the instantaneous market turbulence, rather than imposing a fixed norm across all market regimes.

Remark 4.

Under Assumption 2, the dual of $L^{p(·)}$ is characterized by the linear isomorphism $L^{p^{\prime }(·)}(\Omega ,E^{*})\ni g\mapsto V_g\in {\left(L^{p(·)}\right)}^{*}$ as follows:

$$\langle V_g,f\rangle ={\int }_{\Omega }\langle g,f\rangle d\mu ,\mathit{for}\mathit{any}f\in L^{p(·)}.$$

See Theorem 2 of Cheng and Xu [20].

Example 1.

The variable exponent $p(·)$ in the risk position space is state dependent and can be seen as a response made by investors or market regulators to different market conditions. Consider the space $L^{p(·)}(\Omega ,\mathbb{R})$ on the two-state space $\Omega =\{{\omega }_1,{\omega }_2\}$ with $\mu \left(\left\{{\omega }_1\right\}\right)+\mu \left(\left\{{\omega }_2\right\}\right)=1$ and $\mu \left(\left\{{\omega }_i\right\}\right)>0$, $i=1,2$. The variable exponent $p(·):\Omega \to (1,\infty )$ is given by

$$p\left(\omega \right)=a{\mathbf{1}}_{\left\{{\omega }_1\right\}}\left(\omega \right)+b{\mathbf{1}}_{\left\{{\omega }_2\right\}}\left(\omega \right),\omega \in \Omega ,$$

where ${\mathbf{1}}_A$ denotes the indicator function of a set A and $a,b\in (1,\infty )$, so that $1<p^{-}\le p^{+}<\infty$ in line with Assumption 2. The Bochner–Lebesgue space $L^{p(·)}(\Omega ,\mathbb{R})$ then consists of all random variables on Ω: when the market is in state ${\omega }_1$, the risk position is measured with the $L^a$-geometry, and when it is in state ${\omega }_2$, with the $L^b$-geometry.

The principal contribution of Cheng and Xu [20] is the characterization of the dual of the variable-exponent Bochner–Lebesgue space (see Remark 4) together with its associated properties. While duality is central to functional analysis, it is also fundamental to risk quantification, as it underlies the construction of dual representations of risk measures. Building upon this dual structure, we develop a systemic risk framework and derive explicit dual representations through novel and carefully designed procedures.

3. The Definition of Systemic Risk Measures

In this section, we introduce the axiomatic framework for systemic risk measures on $L^{p(·)}$. Throughout this paper, an element $f\in L^{p(·)}$ represents a loss vector (or risk exposure) of the financial system. Consequently, the monotonicity axioms are interpreted in the natural sense that larger losses entail higher risk. Here, we adopt an axiomatic framework to define systemic risk measures in $L^{p(·)}$. For preparation, we define two special functions.

Definition 4.

Recall that E denotes a Banach space (see Section 2). A convex deterministic function is a mapping $\varphi :E\to \mathbb{R}$ that satisfies the following properties:

A0 

Surjectivity: $\varphi \left(E\right)=\mathfrak{L}$ with $\mathfrak{L}=\mathbb{R}$ or $\mathfrak{L}={\mathbb{R}}_{+}$;

A1 

Monotonicity: for any $x,y\in E$, $x{\ge }_{K}y$ implies $\varphi \left(x\right)\ge \varphi \left(y\right)$;

A2 

Convexity: for any $x,y\in E$ and $\lambda \in [0,1]$, $\varphi (\lambda x+(1-\lambda \left)y\right)\le \lambda \varphi \left(x\right)+(1-\lambda )\varphi \left(y\right)$;

A3 

There exists a variable exponent $r(·)\in S(\Omega ,\mu )$ with $1<r^{-}\le r^{+}<\infty$ (where $r^{-}$ and $r^{+}$ are defined analogously to $p^{-}$ and $p^{+}$), such that

$$\left\{f_{\varphi }:\Omega \ni \omega \mapsto \varphi \left(f\left(\omega \right)\right)\in \mathbb{R}|f\in L^{p(·)}\right\}=L^{r(·)}(\Omega ,\mathbb{R})\mathit{for}\mathfrak{L}=\mathbb{R}$$

$$\left(\mathit{respectively}\left\{f_{\varphi }:\Omega \ni \omega \mapsto \varphi \left(f\left(\omega \right)\right)\in {\mathbb{R}}_{+}|f\in L^{p(·)}\right\}=L^{r(·)}{(\Omega ,\mathbb{R})}_{+}\mathit{for}\mathfrak{L}={\mathbb{R}}_{+}\right).$$

Remark 5.

The order of $A1$ is the partial order induced by the cone K as defined in Remark 1. Thus, the Banach space E is partially ordered by the given cone K. $A0$ states that the function ϕ is non-constant and unbounded from above.

For the range condition A3, we need the following standing assumption on the cone.

Assumption 3.

The cone K is closed and pointed, that is, K is norm closed in E and $K\cap (-K)=\left\{0\right\}$.

Lemma 1.

Let Assumption 3 hold and let $\varphi :E\to \mathbb{R}$ satisfy A0–A2. Suppose that the scalar map $T_{\varphi }:\mathbb{R}\to \mathbb{R}$ defined by $T_{\varphi }\left(a\right):=\varphi \left(az\right)$ is bijective and that there exist constants $\theta \in [1,p^{-})$ and $c_1,c_2>0$ such that

$$c_1\left({\left|a\right|}^{\theta }-1\right)\le \left|T_{\varphi }\left(a\right)\right|\le c_2\left(1+{\left|a\right|}^{\theta }\right),a\in \mathbb{R}.$$

(4)

Then ϕ satisfies A3 with $\mathfrak{L}=\mathbb{R}$, and the range exponent is given by

$$r\left(\omega \right):={\displaystyle \frac{p\left(\omega \right)}{\theta }},\omega \in \Omega ,$$

which satisfies $1<r^{-}\le r^{+}<\infty$.

Proof. 

Note first that $T_{\varphi }$ is non-decreasing by A1, since $a\ge b$ implies $(a-b)z\in K$ and hence $\varphi \left(az\right)\ge \varphi \left(bz\right)$; being bijective, $T_{\varphi }$ is therefore strictly monotone and continuous, and its inverse $T_{\varphi }^{-1}$ is continuous as well.

Since $z\in int\left(K\right)$, there exists $\epsilon >0$ such that $z\pm \epsilon x\in K$ whenever ${\parallel x\parallel }_E\le 1$. For $x\in E$, consider the upper order bound along the numéraire

$$u\left(x\right):=inf\{a\in \mathbb{R}:x{\le }_{K}az\}.$$

For $x\ne 0$, the relation ${(\parallel x\parallel }_E{/\epsilon )z-x=(\parallel x\parallel }_E/\epsilon )\left(z-\epsilon x/{\parallel x\parallel }_E\right)\in K$ shows that the infimum is taken over a non-empty set and that $u\left(x\right)\le {\parallel x\parallel }_E/\epsilon$; the same argument applied to $z+\epsilon x/{\parallel x\parallel }_E\in K$ yields $(\parallel x{\parallel }_E/\epsilon )z+x\in K$. If some $a<-{\parallel x\parallel }_E/\epsilon$ satisfied $az-x\in K$, then with $c:=-(a+\parallel x{\parallel }_E/\epsilon )>0$ we would have $az-x{\le }_K-cz{\le }_{K}0$, because $-cz-(az-x)=(\parallel x{\parallel }_E/\epsilon )z+x\in K$; since K is pointed, this forces $az-x=0$ and $cz\in K\cap (-K)=\left\{0\right\}$, a contradiction. Consequently $\left|u\left(x\right)\right|\le {\parallel x\parallel }_E/\epsilon$ for all $x\in E$. Moreover, if $a_n\downarrow u\left(x\right)$ with $a_{n}z-x\in K$, closedness of K yields $u\left(x\right)z-x\in K$; that is, the infimum is attained: $x{\le }_{K}u\left(x\right)z$ for all $x\in E$. For $x,y\in E$ we have $x-y{\le }_K{(\parallel x-y\parallel }_E/\epsilon )z$ and $y{\le }_{K}u\left(y\right)z$; hence $x{\le }_K\left({u\left(y\right)+\parallel x-y\parallel }_E/\epsilon \right)z$, which gives $u\left(x\right)\le {u\left(y\right)+\parallel x-y\parallel }_E/\epsilon$; interchanging x and y shows that $u:E\to \mathbb{R}$ is Lipschitz continuous with constant $1/\epsilon$. The lower order bound

$$\ell \left(x\right):=sup\{a\in \mathbb{R}:az{\le }_{K}x\}=-u(-x)$$

enjoys the analogous properties: it is Lipschitz continuous with constant $1/\epsilon$, satisfies $|\ell \left(x\right)|\le {\parallel x\parallel }_E/\epsilon$, and the supremum is attained; that is, $\ell \left(x\right)z{\le }_{K}x$ for all $x\in E$.

We also note that $\varphi$ is continuous on E. Indeed, A1 implies $\varphi \left(x\right)\le \varphi \left(z\right)$ whenever ${\parallel x\parallel }_E\le \epsilon$, since then $z-x\in K$; hence the convex function $\varphi$ is bounded above on a neighborhood of the origin, and a convex function on a Banach space that is locally bounded above is continuous.

Now let $f\in L^{p(·)}$. Since $\varphi$ is continuous and f is strongly $\mathcal{F}$-measurable, the set $A:=\{\omega \in \Omega :\varphi (f\left(\omega \right))\ge 0\}$ belongs to $\mathcal{F}$. Define

$$Z\left(\omega \right):=\left\{\begin{array}{cc}u\left(f\right(\omega \left)\right),\hfill & \omega \in A,\hfill \\ \ell \left(f\right(\omega \left)\right),\hfill & \omega \in \Omega \setminus A.\hfill \end{array}\right.$$

Since $A\in \mathcal{F}$ and $u,\ell$ are Lipschitz continuous, Z is $\mathcal{F}$-measurable as the composition of continuous maps with the strongly measurable map f, and $\left|Z\left(\omega \right)\right|\le {\epsilon }^{-1}{\parallel f\left(\omega \right)\parallel }_E$ for all $\omega \in \Omega$, so that $Z\in L^{p(·)}(\Omega ,\mathbb{R})$. For $\omega \in A$, attainment of the infimum gives $f\left(\omega \right){\le }_{K}Z\left(\omega \right)z$, and A1 yields

$$0\le \varphi \left(f\left(\omega \right)\right)\le \varphi \left(Z\left(\omega \right)z\right)=T_{\varphi }\left(Z\left(\omega \right)\right);$$

for $\omega \in \Omega \setminus A$, attainment of the supremum gives $Z\left(\omega \right)z{\le }_{K}f\left(\omega \right)$; hence $T_{\varphi }\left(Z\left(\omega \right)\right)\le \varphi \left(f\left(\omega \right)\right)<0$. In both cases,

$$\left|\varphi \left(f\left(\omega \right)\right)\right|\le |T_{\varphi }\left(Z\left(\omega \right)\right)|\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega .$$

(5)

By the upper bound in (4) and the elementary inequality ${(s+t)}^r\le 2^{r-1}(s^r+t^r)$ for $s,t\ge 0$ and $r\ge 1$,

$${\left|\varphi \left(f\left(\omega \right)\right)\right|}^{r\left(\omega \right)}\le c_2^{r\left(\omega \right)}{\left(1+{\left|Z\left(\omega \right)\right|}^{\theta }\right)}^{r\left(\omega \right)}\le C\left(1+{\left|Z\left(\omega \right)\right|}^{\theta r\left(\omega \right)}\right)=C\left(1+{\left|Z\left(\omega \right)\right|}^{p\left(\omega \right)}\right)$$

for $\mu$-a.e. $\omega \in \Omega$, with a constant $C>0$ depending only on $c_2$ and $r^{+}$. Since ${\left|Z\left(\omega \right)\right|}^{p\left(\omega \right)}\le max\{1,{\epsilon }^{-p^{+}}\}{\parallel f\left(\omega \right)\parallel }_E^{p\left(\omega \right)}$ and $f\in L^{p(·)}$, integrating this inequality gives $\varphi \left(f\right)\in L^{r(·)}(\Omega ,\mathbb{R})$, which proves the inclusion $\{f_{\varphi }:\Omega \ni \omega \mapsto \varphi \left(f\left(\omega \right)\right)|f\in L^{p(·)}\}\subseteq L^{r(·)}(\Omega ,\mathbb{R})$.

Conversely, let $X\in L^{r(·)}(\Omega ,\mathbb{R})$ and set $Y\left(\omega \right):=T_{\varphi }^{-1}\left(X\left(\omega \right)\right)$, which is $\mathcal{F}$-measurable as $T_{\varphi }^{-1}$ is continuous. The lower bound in (4) gives ${\left|a\right|}^{\theta }\le 1+c_1^{-1}\left|T_{\varphi }\left(a\right)\right|$ for all $a\in \mathbb{R}$; hence

$$|T_{\varphi }^{-1}\left(b\right)|\le {\left(1+c_1^{-1}\left|b\right|\right)}^{1/\theta }\le C^{\prime }\left(1+{\left|b\right|}^{1/\theta }\right),b\in \mathbb{R},$$

for a constant $C^{\prime }>0$ depending only on $c_1$ and $\theta$. Therefore

$${\left|Y\left(\omega \right)\right|}^{p\left(\omega \right)}\le C^{\prime \prime }\left(1+{\left|X\left(\omega \right)\right|}^{p\left(\omega \right)/\theta }\right)=C^{\prime \prime }\left(1+{\left|X\left(\omega \right)\right|}^{r\left(\omega \right)}\right),$$

which is integrable since $X\in L^{r(·)}(\Omega ,\mathbb{R})$; hence $Y\in L^{p(·)}(\Omega ,\mathbb{R})$. Setting $f\left(\omega \right):=Y\left(\omega \right)z$, the map f is strongly $\mathcal{F}$-measurable and ${\parallel f\left(\omega \right)\parallel }_E={\left|Y\left(\omega \right)\right|\parallel z\parallel }_E$, so $f\in L^{p(·)}(\Omega ,E)$, and

$$\varphi \left(f\left(\omega \right)\right)=\varphi \left(Y\left(\omega \right)z\right)=T_{\varphi }\left(Y\left(\omega \right)\right)=X\left(\omega \right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega .$$

Thus $L^{r(·)}(\Omega ,\mathbb{R})\subseteq \{f_{\varphi }:\Omega \ni \omega \mapsto \varphi \left(f\left(\omega \right)\right)|f\in L^{p(·)}\}$, and the proof is complete. □

Remark 6.

Under the hypotheses of Lemma 1, condition (4) explains why a fixed-exponent range is generally incompatible with a genuinely variable exponent on a non-atomic state space. Assume, in addition to Assumption 2, that $(\Omega ,\mathcal{F},\mu )$ is non-atomic and that the induced range is required to coincide, as a set, with $L^q(\Omega ,\mathbb{R})$ for some $q\in [1,\infty )$. For every $Z\in L^{p(·)}(\Omega ,{\mathbb{R}}_{+})$, put $f=Zz$. Then $f\in L^{p(·)}(\Omega ,E)$ and $\phi \left(f\right)=T_{\phi }\left(Z\right)$. The lower bound $c_1{\left(\right|Z|}^{\theta }-1)\le |T_{\phi }\left(Z\right)|$ in (4) gives

$${\left|Z\right|}^{\theta }\le 1+c_1^{-1}\left|T_{\phi }\left(Z\right)\right|.$$

Since $T_{\phi }\left(Z\right)\in L^q(\Omega ,\mathbb{R})$, it follows that $Z\in L^{\theta q}(\Omega ,\mathbb{R})$. Hence

$$L^{p(·)}(\Omega ,\mathbb{R})\subseteq L^{\theta q}(\Omega ,\mathbb{R}).$$

On a finite non-atomic measure space, this inclusion is equivalent to $\theta q\le p^{-}$.

Conversely, let $X\in L^q(\Omega ,\mathbb{R})$. By the assumed range equality, there exists $f\in L^{p(·)}(\Omega ,E)$ such that $\phi \left(f\right)=X$. With $u\left(f\right)$ and $\ell \left(f\right)$ as in the proof of Lemma 1, the order bounds $\ell \left(f\right)z\le f\le u\left(f\right)z$ and the monotonicity of $T_{\phi }$ imply

$$\ell \left(f\right)\le T_{\phi }^{-1}\left(X\right)\le u\left(f\right).$$

Since $|\ell (f\left)\right|$ and $\left|u\right(f\left)\right|$ are dominated by a constant multiple of ${\parallel f\parallel }_E$, the function $Y:=T_{\phi }^{-1}\left(X\right)$ belongs to $L^{p(·)}(\Omega ,\mathbb{R})$. Moreover, the upper bound in (4) yields

$${\left|Y\right|}^{\theta }\ge {\displaystyle \frac{\left|X\right|}{2c_2}}\mathit{whenever}\left|X\right|>2c_2,$$

and hence $X\in L^{p(·)/\theta }(\Omega ,\mathbb{R})$. Therefore

$$L^q(\Omega ,\mathbb{R})\subseteq L^{p(·)/\theta }(\Omega ,\mathbb{R}),$$

which, again by non-atomicity, is equivalent to $\theta q\ge p^{+}$. Thus $p^{+}\le \theta q\le p^{-}$, forcing $p^{+}=p^{-}$ and hence $p(·)=\theta q$μ-almost everywhere. Consequently, on a non-atomic probability space, a genuinely variable exponent requires the variable range exponent $r(·)=p(·)/\theta$; in the constant case $p(·)\equiv p$, the fixed range exponent $q=p/\theta$ is recovered.

The preceding obstruction relies on non-atomicity and does not apply to finite atomic models. If $\Omega =\{{\omega }_1,\dots ,{\omega }_m\}$ and every atom has positive probability then every measurable function on Ω is bounded. Consequently,

$$L^{p(·)}(\Omega ,\mathbb{R})=L^q(\Omega ,\mathbb{R})$$

as sets for every bounded exponent $p(·)$ and every $q\in [1,\infty )$, although the corresponding norms and modulars need not coincide. Hence fixed-exponent ranges in finite-regime models, such as those in Section 6, are not excluded by the non-atomic obstruction above. In those examples, the fixed exponent is simply a convenient representative of the common underlying set of functions; it does not imply that $p(·)$ is constant.

In fact, the convex deterministic function $\varphi$ is used to transform the uncertainty of systemic risk into certainty. However, to measure systemic risk in $L^{p(·)}$, we still require a convex single-firm risk measure to quantify the risk simplified by $\varphi$.

Definition 5.

A convex single-firm risk measure on $L^{r(·)}(\Omega ,\mathbb{R})$, where $r(·)\in S(\Omega ,\mu )$ satisfies $1<r^{-}\le r^{+}<\infty$, is a functional

$$\varrho :L^{r(·)}(\Omega ,\mathbb{R})\ni f\mapsto \varrho \left(f\right)\in \mathbb{R}\cup \{+\infty \}$$

that satisfies the following properties: B1 Monotonicity: for any $X,Y\in L^{r(·)}(\Omega ,\mathbb{R})$, $X\ge Y$ implies $\varrho \left(X\right)\ge \varrho \left(Y\right)$; B2 Convexity: for any $X,Y\in L^{r(·)}(\Omega ,\mathbb{R})$ and $\lambda \in [0,1]$, $\varrho \left(\lambda X+(1-\lambda )Y\right)\le \lambda \varrho \left(X\right)+(1-\lambda )\varrho \left(Y\right)$; B3 Constancy: for any $a\in \mathbb{R}$, $\varrho \left(a\right)=a$.

Remark 7.

$B1$–$B2$ are well known and have been studied in detail in works on convex risk measures (see, for instance, Föllmer and Schied [30]). $B3$ can be understood as a technical condition. Since $(\Omega ,\mathcal{F},\mu )$ is a probability space, all constants belong to $L^{r(·)}(\Omega ,\mathbb{R})$, so B3 is meaningful; the classical fixed-exponent setting on $L^{r(·)}(\Omega ,\mathbb{R})$ is recovered when $r(·)\equiv q$.

Remark 8.

The two mappings in Definitions 4 and 5 divide the measurement of systemic risk into two steps: the deterministic function converts the uncertainty of systemic risk into certainty, and the convex single-firm risk measure quantifies the risk simplified by the deterministic function.

We now introduce the definition of convex systemic risk measures in the variable-exponent Bochner–Lebesgue space $L^{p(·)}$ via an axiomatic approach.

Definition 6.

A convex systemic risk measure is a functional

$$\begin{array}{c}\hfill \rho :L^{p(·)}\ni f\mapsto \rho \left(f\right)\in \mathbb{R}\cup \{+\infty \}\end{array}$$

that satisfies the following properties:

C0 

Surjectivity: $\rho \left(E\right)=\mathfrak{L}$ with $\mathfrak{L}=\mathbb{R}$ or $\mathfrak{L}={\mathbb{R}}_{+}$. Here, E is identified with the set of all constant mappings from Ω to E (or, equivalently, deterministic E-valued payoffs);

C1 

Monotonicity: for any $f,g\in L^{p(·)}$, $f\left(\omega \right){\ge }_{K}g\left(\omega \right)$ implies $\rho \left(f\right)\ge \rho \left(g\right)$;

C2 

Preference consistency: If $\rho \left(f\right(\omega \left)\right)\le \rho \left(g\right(\omega \left)\right)$ for all $\omega \in \Omega$, then $\rho \left(f\right)\le \rho \left(g\right)$;

C3 

Convexity: for any $f,g\in L^{p(·)}$ and $\lambda \in [0,1]$, $\rho (\lambda f+(1-\lambda \left)g\right)\le \lambda \rho \left(f\right)+(1-\lambda )\rho \left(g\right)$;

C4 

Risk convexity: if $\rho \left(h\right(\omega \left)\right)=\lambda \rho \left(f\right(\omega \left)\right)+(1-\lambda )\rho \left(g\right(\omega \left)\right)$ for a given scalar $\lambda \in [0,1]$ and for all $\omega \in \Omega$ then $\rho \left(h\right)\le \lambda \rho \left(f\right)+(1-\lambda )\rho \left(g\right)$;

C5 

There exists a variable exponent $r(·)\in S(\Omega ,\mu )$ with $1<r^{-}\le r^{+}<\infty$, such that

$$\left\{f_{\rho }:\Omega \ni \omega \mapsto \rho \left(f\left(\omega \right)\right)\in \mathbb{R}|f\in L^{p(·)}\right\}=L^{r(·)}(\Omega ,\mathbb{R})\mathit{for}\mathfrak{L}=\mathbb{R}$$

$$\left(\mathit{respectively}\left\{f_{\rho }:\Omega \ni \omega \mapsto \rho \left(f\left(\omega \right)\right)\in {\mathbb{R}}_{+}|f\in L^{p(·)}\right\}=L^{r(·)}{(\Omega ,\mathbb{R})}_{+}\mathit{for}\mathfrak{L}={\mathbb{R}}_{+}\right),$$

where, for each fixed $\omega \in \Omega$, $\rho \left(f\right(\omega \left)\right)$ is understood as the image of the constant mapping (or sure payoff) $\Omega \ni {\omega }^{\prime }\mapsto f\left(\omega \right)\in E$ under the functional ρ.

Remark 9.

$C1$ and $C3$ can be interpreted in the same way as in the definition of single-firm risk measures. $C2$ means that if the economic risk $f\left(\omega \right)\in E$ is greater than the economic risk $g\left(\omega \right)\in E$ for almost all $\omega \in \Omega$ then the random economic risk of $f\in L^{p(·)}$ should be greater than that of $g\in L^{p(·)}$. $C4$ states that if the economic risk $h\left(\omega \right)$ is a convex combination of the risks of the economies $f\left(\omega \right)$ and $g\left(\omega \right)$ for all $\omega \in \Omega$ then the random economic risk of the economy $h\in L^{p(·)}$ is at most that of the convex combination of the random economies $f,g\in L^{p(·)}$. $C5$ is a technical requirement. $C0$ and $A0$ of the corresponding functions are closely linked, and we will use these properties for our decomposition of the measurements in the following section; specifically, the condition $\varphi \left(E\right)=\mathbb{R}=\rho \left(E\right)$ is needed.

Remark 10.

The two alternatives $\mathfrak{L}=\mathbb{R}$ and $\mathfrak{L}={\mathbb{R}}_{+}$ in A0, C0, A3 and C5 play different roles in the subsequent decomposition results. In the full-range branch $\mathfrak{L}=\mathbb{R}$, A3 identifies the induced range with the whole space $L^{r(·)}(\Omega ,\mathbb{R})$. Consequently, the outer single-firm risk measure constructed in Section 4 is defined on the full domain required by Definition 5.

By contrast, in the positive-range branch $\mathfrak{L}={\mathbb{R}}_{+}$, A3 identifies the induced range only with the positive cone $L_{+}^{r(·)}(\Omega ,\mathbb{R})$. We therefore do not claim a decomposition equivalence for this branch. Instead, positive-range deterministic functions will be used in direct composition constructions with single-firm risk measures already defined on the whole space $L^{r(·)}(\Omega ,\mathbb{R})$.

Definition 7.

(1) A coherent deterministic function is a function $\varphi :E\to \mathbb{R}$ with $\left\{f_{\varphi }:\Omega \ni \omega \mapsto \varphi \left(f\left(\omega \right)\right)\in \mathbb{R}|f\in L^{p(·)}\right\}$$=L^{r(·)}(\Omega ,\mathbb{R})$ (respectively $=L^{r(·)}{(\Omega ,\mathbb{R})}_{+}$) for a variable exponent $r(·)\in S(\Omega ,\mu )$ with $1<r^{-}\le r^{+}<\infty$ that satisfies $A0$–$A2$ and has the following properties:

A4 

Positive homogeneity: for any $x\in E$ and $t\in {\mathbb{R}}_{+}$, $\varphi \left(tx\right)=t\varphi \left(x\right)$;

A5 

Normalization: $\varphi \left(z\right)=1$.

(2) A coherent single-firm risk measure is a function $\varrho :$$L^{r(·)}(\Omega ,\mathbb{R})\to \mathbb{R}\cup \{+\infty \}$ that satisfies $B1$–$B3$ and the following properties:

B4 

Positive homogeneity: for any $X\in L^{r(·)}(\Omega ,\mathbb{R})$ and $t\in {\mathbb{R}}_{+}$, $\varrho \left(tX\right)=t\varrho \left(X\right)$;

B5 

Normalization: $\varrho \left(1\right)=1$.

(3) A coherent systemic risk measure is a function ρ: $L^{p(·)}$→$\mathbb{R}\cup \{+\infty \}$ with $\left\{f_{\rho }:\Omega \ni \omega \mapsto \rho \left(f\left(\omega \right)\right)\in \mathbb{R}|f\in L^{p(·)}\right\}$$=L^{r(·)}(\Omega ,\mathbb{R})$ (respectively $=L^{r(·)}{(\Omega ,\mathbb{R})}_{+}$) for a variable exponent $r(·)\in S(\Omega ,\mu )$ with $1<r^{-}\le r^{+}<\infty$ that satisfies $C0$–$C4$ and the following properties:

C6 

Positive homogeneity: for any $f\in L^{p(·)}$ and $t\in {\mathbb{R}}_{+}$, $\rho \left(tf\right)=t\rho \left(f\right)$;

C7 

Normalization: $\rho \left(z\right)=1$.

In the upcoming Section 4, we will demonstrate how to compose a deterministic function with a single-firm risk measure to complete the construction of systemic risk measures under market volatility.

4. Systemic Risk Measures on ${\mathit{L}}^{\mathit{p}\mathbf{(}\mathbf{·}\mathbf{)}}$

In this section, we investigate systemic risk quantification on the variable-exponent Bochner–Lebesgue space $L^{p(·)}$ and provide a structural decomposition result showing that each systemic risk measure in $L^{p(·)}$ can be decomposed into a deterministic function and a single-firm risk measure. Furthermore, we show that any convex deterministic function and convex single-firm risk measure can be aggregated into a systemic risk measure in $L^{p(·)}$.

Theorem 1.

A functional $\rho :L^{p(·)}\to \mathbb{R}\cup \{+\infty \}$ is a convex systemic risk measure with the full-range branch $\mathfrak{L}=\mathbb{R}$ in C0 if and only if there exist a variable exponent $r(·)\in \mathcal{S}(\Omega ,\mu )$ with $1<r^{-}\le r^{+}<\infty$, a convex deterministic function $\varphi :E\to \mathbb{R}$ satisfying A0 with $\mathfrak{L}=\mathbb{R}$, and a convex single-firm risk measure

$$\varrho :L^{r(·)}(\Omega ,\mathbb{R})\to \mathbb{R}\cup \{+\infty \}$$

such that

$$\rho \left(f\right)=(\varrho \circ \varphi )\left(f\right),f\in L^{p(·)}.$$

(6)

Proof. 

We first prove necessity. Suppose that $\rho$ is a convex systemic risk measure with $\mathfrak{L}=\mathbb{R}$, and define

$$\varphi \left(x\right):=\rho \left(x\right),x\in E.$$

(7)

Since $\rho$ satisfies C0 with $\mathfrak{L}=\mathbb{R}$, the function $\varphi$ satisfies A0 with the same branch. A1 follows from C1, and A2 follows from C3. Moreover, since

$$\rho \left(f\right(\omega \left)\right)=\varphi \left(f\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega ,$$

C5 of $\rho$ implies A3 of $\varphi$ with the same full-range exponent $r(·)$. Hence $\varphi$ is a convex deterministic function with $\varphi \left(E\right)=\mathbb{R}$.

By A3, for every $X\in L^{r(·)}(\Omega ,\mathbb{R})$ there exists $f\in L^{p(·)}$ such that $\varphi \left(f\right)=X$. We can therefore define

$$\varrho \left(X\right):=\rho \left(f\right),\varphi \left(f\right)=X.$$

(8)

To see that $\varrho$ is well defined, let $f,g\in L^{p(·)}$ satisfy $\varphi \left(f\right)=\varphi \left(g\right)$. Then

$$\rho \left(f\right(\omega \left)\right)=\varphi \left(f\right(\omega \left)\right)=\varphi \left(g\right(\omega \left)\right)=\rho \left(g\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega .$$

To see that $\varrho$ is well defined, let $f,g\in L^{p(·)}$ satisfy $\varphi \left(f\right)=\varphi \left(g\right)$. By the definition (7),

$$\rho \left(f\right(\omega \left)\right)=\varphi \left(f\right(\omega \left)\right)=\varphi \left(g\right(\omega \left)\right)=\rho \left(g\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega .$$

Applying C2 to the pair $(f,g)$ yields $\rho \left(f\right)\le \rho \left(g\right)$, and applying C2 to the pair $(g,f)$ yields $\rho \left(g\right)\le \rho \left(f\right)$. Hence the two global risk values coincide, $\rho \left(f\right)=\rho \left(g\right)$, so the value in (8) does not depend on the chosen pre-image. Thus the value in (8) does not depend on the chosen pre-image. Since A3 gives

$$\{\varphi \left(f\right):f\in L^{p(·)}\}=L^{r(·)}(\Omega ,\mathbb{R}),$$

the functional $\varrho$ is defined on the whole space $L^{r(·)}(\Omega ,\mathbb{R})$.

We next verify B1–B3. Let $X,Y\in L^{r(·)}(\Omega ,\mathbb{R})$ satisfy $X\ge Y$. Choose $f,g\in L^{p(·)}$ such that $\varphi \left(f\right)=X$ and $\varphi \left(g\right)=Y$. Then

$$\rho \left(f\right(\omega \left)\right)=\varphi \left(f\right(\omega \left)\right)\ge \varphi \left(g\right(\omega \left)\right)=\rho \left(g\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega .$$

By C2,

$$\varrho \left(X\right)=\rho \left(f\right)\ge \rho \left(g\right)=\varrho \left(Y\right),$$

which proves B1.

For B2, let $X,Y\in L^{r(·)}(\Omega ,\mathbb{R})$, $\lambda \in [0,1]$, and set

$$Z:=\lambda X+(1-\lambda )Y.$$

By A3, there exist $f,g,h\in L^{p(·)}$ such that

$$\varphi \left(f\right)=X,\varphi \left(g\right)=Y,\varphi \left(h\right)=Z.$$

Then

$$\begin{array}{cc}\hfill \rho \left(h\right(\omega \left)\right)& =\varphi \left(h\right(\omega \left)\right)=Z\left(\omega \right)\hfill \\ & =\lambda X\left(\omega \right)+(1-\lambda )Y\left(\omega \right)\hfill \\ & =\lambda \varphi \left(f\right(\omega \left)\right)+(1-\lambda )\varphi \left(g\right(\omega \left)\right)\hfill \\ & =\lambda \rho \left(f\right(\omega \left)\right)+(1-\lambda )\rho \left(g\right(\omega \left)\right)\hfill \end{array}$$

for $\mu$-a.e. $\omega \in \Omega$. C4 therefore gives

$$\varrho \left(Z\right)=\rho \left(h\right)\le \lambda \rho \left(f\right)+(1-\lambda )\rho \left(g\right)=\lambda \varrho \left(X\right)+(1-\lambda )\varrho \left(Y\right),$$

which proves B2.

Finally, for every $a\in \mathbb{R}$, A0 gives $x\in E$ such that $\varphi \left(x\right)=a$. It follows from (7) and (8) that

$$\varrho \left(a\right)=\rho \left(x\right)=\varphi \left(x\right)=a.$$

Thus B3 holds, and $\varrho$ is a convex single-firm risk measure on the whole space $L^{r(·)}(\Omega ,\mathbb{R})$. By construction,

$$\rho \left(f\right)=\varrho \left(\varphi \left(f\right)\right),f\in L^{p(·)}.$$

We now prove sufficiency. Let $\varphi$ be a convex deterministic function with $\varphi \left(E\right)=\mathbb{R}$, let $\varrho$ be a convex single-firm risk measure on $L^{r(·)}(\Omega ,\mathbb{R})$, and define

$$\rho :=\varrho \circ \varphi .$$

For every $x\in E$, B3 gives

$$\rho \left(x\right)=\varrho \left(\varphi \right(x\left)\right)=\varphi \left(x\right).$$

Consequently,

$$\rho \left(E\right)=\varphi \left(E\right)=\mathbb{R},$$

which proves C0 with $\mathfrak{L}=\mathbb{R}$.

If $f{\ge }_{K}g$ then A1 gives $\varphi \left(f\right)\ge \varphi \left(g\right)$. B1 therefore implies

$$\rho \left(f\right)=\varrho \left(\varphi \right(f\left)\right)\ge \varrho \left(\varphi \right(g\left)\right)=\rho \left(g\right),$$

which proves C1.

For C2, suppose that

$$\rho \left(f\right(\omega \left)\right)\ge \rho \left(g\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega .$$

Since $\varphi \left(f\right(\omega \left)\right)$ is a real constant for each fixed $\omega$, B3 yields

$$\rho \left(f\right(\omega \left)\right)=\varrho \left(\varphi \right(f\left(\omega \right)\left)\right)=\varphi \left(f\right(\omega \left)\right).$$

The same relation holds for g. Hence

$$\varphi \left(f\right(\omega \left)\right)\ge \varphi \left(g\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega ,$$

and B1 gives

$$\rho \left(f\right)=\varrho \left(\varphi \right(f\left)\right)\ge \varrho \left(\varphi \right(g\left)\right)=\rho \left(g\right).$$

Convexity C3 follows from A2, B1 and B2:

$$\begin{array}{cc}\hfill \rho (\lambda f+(1-\lambda \left)g\right)& =\varrho \left(\varphi (\lambda f+(1-\lambda \left)g\right)\right)\hfill \\ & \le \varrho \left(\lambda \varphi \left(f\right)+(1-\lambda )\varphi \left(g\right)\right)\hfill \\ & \le \lambda \varrho \left(\varphi \right(f\left)\right)+(1-\lambda )\varrho \left(\varphi \right(g\left)\right)\hfill \\ & =\lambda \rho \left(f\right)+(1-\lambda )\rho \left(g\right).\hfill \end{array}$$

For C4, suppose that

$$\rho \left(h\right(\omega \left)\right)=\lambda \rho \left(f\right(\omega \left)\right)+(1-\lambda )\rho \left(g\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega .$$

Using B3 as above, this is equivalent to

$$\varphi \left(h\right)=\lambda \varphi \left(f\right)+(1-\lambda )\varphi \left(g\right).$$

Therefore B2 gives

$$\begin{array}{cc}\hfill \rho \left(h\right)& =\varrho \left(\varphi \left(h\right)\right)=\varrho \left(\lambda \varphi \left(f\right)+(1-\lambda )\varphi \left(g\right)\right)\hfill \\ & \le \lambda \varrho \left(\varphi \right(f\left)\right)+(1-\lambda )\varrho \left(\varphi \right(g\left)\right)=\lambda \rho \left(f\right)+(1-\lambda )\rho \left(g\right).\hfill \end{array}$$

Finally,

$$\rho \left(f\right(\omega \left)\right)=\varphi \left(f\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega ,$$

so C5 follows from the full-range branch of A3. Hence $\rho$ is a convex systemic risk measure with $\mathfrak{L}=\mathbb{R}$. □

Remark 11.

Theorem 1 not only offers a decomposition result for convex systemic risk measures in $L^{p(·)}$ but also introduces a framework for addressing systemic risk in markets characterized by uncertainty and volatility. Specifically, we first apply the convex deterministic function ϕ to transform the uncertainty of systemic risk into a certain quantity, and then we quantify this simplified risk using a convex single-firm risk measure. As a result, a regulator responsible for measuring systemic risk can develop a reasonable systemic risk measure by selecting an appropriate deterministic function and a suitable single-firm risk measure. The deterministic function should capture the relevant preferences concerning the uncertainty and volatility of the financial market.

The following theorem concerns coherent systemic risk measures on the variable-exponent Bochner–Lebesgue space $L^{p(·)}$. The decomposition parallels that of Theorem 1, with the two component functions now provided by the coherent deterministic function and the coherent single-firm risk measure introduced in Definition 7.

Theorem 2.

A functional $\rho :L^{p(·)}\to \mathbb{R}\cup \{+\infty \}$ is a coherent systemic risk measure with the full-range branch $\mathfrak{L}=\mathbb{R}$ in C0 if and only if there exist a coherent deterministic function $\varphi :E\to \mathbb{R}$ satisfying $\varphi \left(E\right)=\mathbb{R}$, and a coherent single-firm risk measure

$$\varrho :L^{r(·)}(\Omega ,\mathbb{R})\to \mathbb{R}\cup \{+\infty \}$$

such that

$$\rho \left(f\right)=(\varrho \circ \varphi )\left(f\right),f\in L^{p(·)},$$

(9)

where $r(·)$ is the range exponent associated with ϕ.

Proof. 

Suppose first that $\rho$ is a coherent systemic risk measure with $\mathfrak{L}=\mathbb{R}$. In particular, $\rho$ is a convex systemic risk measure with the same full-range branch. By Theorem 1, there exist a convex deterministic function $\varphi$ satisfying $\varphi \left(E\right)=\mathbb{R}$ and a convex single-firm risk measure $\varrho$ on $L^{r(·)}(\Omega ,\mathbb{R})$ such that

$$\rho =\varrho \circ \varphi .$$

For every $x\in E$ and $t\ge 0$, C6 gives

$$\varphi \left(tx\right)=\rho \left(tx\right)=t\rho \left(x\right)=t\varphi \left(x\right),$$

which proves A4. Moreover, C7 gives

$$\varphi \left(z\right)=\rho \left(z\right)=1,$$

which proves A5. Hence $\varphi$ is a coherent deterministic function with full range $\mathbb{R}$.

It remains to verify B4 and B5 for $\varrho$. Let $X\in L^{r(·)}(\Omega ,\mathbb{R})$. By A3, there exists $f\in L^{p(·)}$ such that $\varphi \left(f\right)=X$. For every $t\ge 0$, A4 gives

$$\varphi \left(tf\right)=t\varphi \left(f\right)=tX.$$

Using C6, we obtain

$$\varrho \left(tX\right)=\rho \left(tf\right)=t\rho \left(f\right)=t\varrho \left(X\right),$$

which proves B4. Finally,

$$\varrho \left(1\right)=\rho \left(z\right)=1,$$

so B5 holds. Thus $\varrho$ is a coherent single-firm risk measure.

Conversely, let $\varphi$ be a coherent deterministic function with $\varphi \left(E\right)=\mathbb{R}$, and let $\varrho$ be a coherent single-firm risk measure on $L^{r(·)}(\Omega ,\mathbb{R})$. By the sufficiency part of Theorem 1, $\rho =\varrho \circ \varphi$ is a convex systemic risk measure with $\mathfrak{L}=\mathbb{R}$. For every $f\in L^{p(·)}$ and $t\ge 0$, A4 and B4 yield

$$\rho \left(tf\right)=\varrho \left(\varphi \right(tf\left)\right)=\varrho \left(t\varphi \right(f\left)\right)=t\varrho \left(\varphi \right(f\left)\right)=t\rho \left(f\right),$$

which proves C6. Moreover, A5 and B5 give

$$\rho \left(z\right)=\varrho \left(\varphi \right(z\left)\right)=\varrho \left(1\right)=1,$$

which proves C7. Hence $\rho$ is a coherent systemic risk measure. □

The first part of the proof of Theorem 2 directly invokes the conclusions of Theorem 1, since most of the axiomatic properties required for coherent systemic risk measures coincide with those of convex systemic risk measures. Nevertheless, the imposition of positive homogeneity carries distinct implications, as detailed in Chen et al. [1].

Proposition 1

(Positive-range direct construction). Let $\varphi :E\to \mathbb{R}$ satisfy A1–A3 and the positive-range branch

$$\varphi \left(E\right)={\mathbb{R}}_{+}$$

of A0. Let $r(·)$ be the associated range exponent in A3, and let ϱ be a convex single-firm risk measure defined on the whole space $L^{r(·)}(\Omega ,\mathbb{R})$. Then

$$\rho :=\varrho \circ \varphi$$

is a convex systemic risk measure with the positive-range branch $\mathfrak{L}={\mathbb{R}}_{+}$ in C0 and C5.

If, in addition, ϕ satisfies A4–A5 and ϱ satisfies B4–B5 then ρ is a coherent systemic risk measure with the same positive-range branch.

Proof. 

For every $x\in E$, B3 gives

$$\rho \left(x\right)=\varrho \left(\varphi \right(x\left)\right)=\varphi \left(x\right).$$

Since $\varphi \left(E\right)={\mathbb{R}}_{+}$, it follows that

$$\rho \left(E\right)={\mathbb{R}}_{+},$$

which is C0 with $\mathfrak{L}={\mathbb{R}}_{+}$. Similarly, for every $f\in L^{p(·)}$,

$$\rho \left(f\right(\omega \left)\right)=\varrho \left(\varphi \right(f\left(\omega \right)\left)\right)=\varphi \left(f\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega .$$

Therefore the positive-range branch of C5 follows directly from the positive-range branch of A3.

Properties C1 and C3 follow from A1–A2 together with B1–B2. For C2, if

$$\rho \left(f\right(\omega \left)\right)\ge \rho \left(g\right(\omega \left)\right)\mathrm{for}\mu -\mathrm{a}.\mathrm{e}.\omega \in \Omega$$

then B3 implies

$$\varphi \left(f\right(\omega \left)\right)\ge \varphi \left(g\right(\omega \left)\right).$$

Applying B1 gives $\rho \left(f\right)\ge \rho \left(g\right)$. The same argument, combined with B2, proves C4. Hence $\rho$ is a convex systemic risk measure with positive range.

Under the additional coherence assumptions, A4 and B4 imply

$$\rho \left(tf\right)=\varrho \left(\varphi \right(tf\left)\right)=\varrho \left(t\varphi \right(f\left)\right)=t\rho \left(f\right),t\ge 0,$$

while A5 and B5 imply

$$\rho \left(z\right)=\varrho \left(\varphi \right(z\left)\right)=\varrho \left(1\right)=1.$$

Thus C6 and C7 hold, and $\rho$ is coherent. □

Remark 12.

Proposition 1 is only a direct construction result. It does not assert a decomposition equivalence in the positive-range branch. Indeed, if the range of ϕ is only $L_{+}^{r(·)}(\Omega ,\mathbb{R})$ them the formula $\varrho \left(\varphi \right(f\left)\right)=\rho \left(f\right)$ determines ϱ merely on that positive cone, whereas Definition 5 requires ϱ to be defined on the whole space $L^{r(·)}(\Omega ,\mathbb{R})$ and to satisfy $\varrho \left(a\right)=a$ for every $a\in \mathbb{R}$.

In the following section, we derive the dual representations of systemic risk measures in $L^{p(·)}$ in terms of the acceptance sets of $\varphi$ and $\varrho$.

5. Dual Representations of Systemic Risk Measures in ${\mathit{L}}^{\mathit{p}\mathbf{(}\mathbf{·}\mathbf{)}}$

Although the preceding section achieved a structured characterization of systemic risk positions through the two-stage decomposition involving a deterministic function and a single-firm risk measure, the resulting construction remains purely forward-looking. It prescribes how complex multi-institutional exposures are mapped into measurable variables, yet it furnishes no computable theoretical framework. In other words, knowing how to construct is insufficient to answer the central question of how to quantify. Consequently, dual representations of the resulting systemic risk measure are needed.

Dual representations confer several decisive advantages for risk quantification. By transforming the originally intractable primal optimization problem into a computable convex conjugate formulation they render regulatory parameters such as liquidity quotas operationally implementable. In addition, the dual variables explicitly quantify the marginal contribution of each institution to aggregate risk. Moreover, embedding the risk measure within the convex analytic framework furnished by duality supplies a unified apparatus for subsequent sensitivity analysis, robustness testing, and model calibration.

Throughout this section, $r(·)\in S(\Omega ,\mu )$ denotes the range exponent associated with $\varphi$ as in Lemma 1, and $r^{\prime }(·)\in S(\Omega ,\mu )$ denotes its pointwise conjugate exponent, $1/r\left(\omega \right)+1/r^{\prime }\left(\omega \right)=1$ for all $\omega \in \Omega$. Since $r^{-}>1$, the space $L^{r(·)}(\Omega ,\mathbb{R})$ is reflexive and its dual is identified with $L^{r^{\prime }(·)}$ via Remark 4. Accordingly, closedness, lower semicontinuity and continuity below refer to these norm topologies, and all conjugates are taken with respect to these pairings. For a set $\mathcal{A}$, we write ${\mathbb{I}}_{\mathcal{A}}$ for its indicator function, which takes the value 0 on $\mathcal{A}$ and $+\infty$ otherwise, and ${\mathbb{I}}_{\mathcal{A}}^{*}$ for its conjugate, i.e., the support function of $\mathcal{A}$.

Remark 13.

Under the assumptions of Lemma 1, the map $L^{p(·)}(\Omega ,E)\ni f\mapsto \varphi \left(f\right)\in L^{r(·)}(\Omega ,\mathbb{R})$ is continuous. Indeed, if $f_n\to f$ in $L^{p(·)}$ then every subsequence of $\left(f_n\right)$ has a further subsequence converging μ-a.e. to f with convergent modulars; continuity of ϕ on E gives $\varphi \left(f_n\left(\omega \right)\right)\to \varphi \left(f\left(\omega \right)\right)$μ-a.e., and the bound

$$|\varphi \left(f_n\left(\omega \right)\right){|}^{r\left(\omega \right)}\le C\left(1+\parallel f_n{\left(\omega \right)\parallel }_E^{p\left(\omega \right)}\right)$$

obtained from (5), the upper bound in (4), and the estimate $\left|Z\left(\omega \right)\right|\le {\epsilon }^{-1}{\parallel f\left(\omega \right)\parallel }_E$ established in the proof of Lemma 3.1, yields uniform integrability. Vitali’s convergence theorem then gives $\varphi \left(f_n\right)\to \varphi \left(f\right)$ in $L^{r(·)}(\Omega ,\mathbb{R})$. In particular, the acceptance set ${\mathcal{A}}_{\varphi }$ below is a closed convex subset of $L^{r(·)}(\Omega ,\mathbb{R})\times L^{p(·)}$ whenever ϕ is continuous, as required in the proof of Theorem 3.

Before studying the dual representations of convex systemic risk measures in $L^{p(·)}$, we first introduce the acceptance sets. Since every systemic risk measure $\rho$ can be decomposed into a convex deterministic function $\varphi$ and a convex single-firm risk measure $\varrho$, we next define the acceptance sets of $\varphi$ and $\varrho$ as follows:

$${\mathcal{A}}_{\varrho }:=\{(c,X)\in \mathbb{R}\times L^{r(·)}(\Omega ,\mathbb{R})|\varrho \left(X\right)\le c\}$$

(10)

and

$${\mathcal{A}}_{\varphi }:=\{(Y,f)\in L^{r(·)}(\Omega ,\mathbb{R})\times L^{p(·)}|\varphi \left(f\right)\le Y\}.$$

(11)

We will see later that these acceptance sets can be used to provide a representation result for systemic risk measures in $L^{p(·)}$. The following properties are required for the subsequent study.

Definition 8.

Let M and N be two ordered linear spaces. A set $A\subset M\times N$ satisfies l-monotonicity if $(m,n)\in A$, $u\in N$ and $n\ge u$ imply $(m,u)\in A$. A set $A\subset M\times N$ satisfies b-monotonicity if $(m,n)\in A$, $v\in M$ and $v\ge m$ imply $(v,n)\in A$.

Proposition 2.

Suppose that $\rho =\varrho \circ \varphi$ is a systemic risk measure with deterministic function $\varphi :E\to \mathbb{R}$ and a single-firm risk measure $\varrho :$$L^{r(·)}(\Omega ,\mathbb{R})\to \mathbb{R}\cup \{+\infty \}$. The corresponding acceptance sets ${\mathcal{A}}_{\varrho }$ and ${\mathcal{A}}_{\varphi }$ are defined by (10) and (11). Then, ${\mathcal{A}}_{\varphi }$ and ${\mathcal{A}}_{\varrho }$ are convex sets and both of them satisfy the l-monotonicity and b-monotonicity.

Proof. 

It is easy to check the above properties by the definitions of $\varphi$ and $\varrho$. □

The following proposition presents the primal representation of systemic risk measures in $L^{p(·)}$ in terms of the acceptance sets. This result serves as the foundation for the representation results that follow.

Proposition 3.

Suppose that $\rho =\varrho \circ \varphi$ is a systemic risk measure with convex deterministic function $\varphi :E\to \mathbb{R}$ and a convex single-firm risk measure $\varrho :$$L^{r(·)}(\Omega ,\mathbb{R})\to \mathbb{R}\cup \{+\infty \}$. The corresponding acceptance sets ${\mathcal{A}}_{\varrho }$ and ${\mathcal{A}}_{\varphi }$ are defined by (10) and (11). Then, for any $f\in L^{p(·)}$,

$$\rho \left(f\right)=inf\left\{c\in \mathbb{R}|(c,X)\in {\mathcal{A}}_{\varrho },(X,f)\in {\mathcal{A}}_{\varphi }\right\}$$

(12)

with $inf\varnothing =\infty$.

Proof. 

As $\rho =\varrho \circ \varphi$, we have

$$\rho \left(f\right)=inf\left\{c\in \mathbb{R}\left|\right(\varrho \circ \varphi \left)\right(f)\le c\right\}.$$

(13)

By the definition of ${\mathcal{A}}_{\varrho }$, we know that

$$\varrho \left(X\right)=inf\left\{c\in \mathbb{R}|(c,X)\in {\mathcal{A}}_{\varrho }\right\}$$

(14)

for all $X\in L^{r(·)}(\Omega ,\mathbb{R})$. Then, from (13) and (14), $\rho \left(f\right)=inf\left\{c\in \mathbb{R}|(c,\varphi \left(f\right))\in {\mathcal{A}}_{\varrho }\right\}.$ It is relatively simple to check that $\left\{c\in \mathbb{R}|(c,\varphi \left(f\right))\in {\mathcal{A}}_{\varrho }\right\}=\left\{c\in \mathbb{R}|(c,X)\in {\mathcal{A}}_{\varrho },(X,f)\in {\mathcal{A}}_{\varphi }\right\}.$ Thus, $\rho \left(f\right)=inf\left\{c\in \mathbb{R}|(c,X)\in {\mathcal{A}}_{\varrho },(X,f)\in {\mathcal{A}}_{\varphi }\right\}.$ □

With the help of Proposition 3, we introduce the main result of this section: the representation result of the convex systemic risk measures in $L^{p(·)}$.

Theorem 3.

Suppose $\rho =\varrho \circ \varphi$, where ϱ is a lower semicontinuous convex single-firm risk measure and ϕ is a continuous convex deterministic function. Then, for any $f\in L^{p(·)}$, $\rho \left(f\right)$ has the following form

$$\rho \left(f\right)=\underset{(Y^{*},f^{*})\in P}{sup}\left\{\langle f^{*},f\rangle -\alpha (Y^{*},f^{*})\right\},$$

(15)

where $\alpha :L^{r^{\prime }(·)}(\Omega ,\mathbb{R})\times {\left(L^{p(·)}\right)}^{*}\to \mathbb{R}\cup \{+\infty \}$ is defined by

$$\alpha (Y^{*},f^{*}):=\underset{(c,X)\in A_{\varrho }}{sup}\underset{(Y,g)\in A_{\varphi }}{sup}\left\{-c-\langle Y^{*},Y-X\rangle +\langle f^{*},g\rangle \right\}$$

(16)

and

$$P:=\left\{(Y^{*},f^{*})\in L^{r^{\prime }(·)}(\Omega ,\mathbb{R})\times {\left(L^{p(·)}\right)}^{*}:\alpha (Y^{*},f^{*})<\infty \right\}.$$

Proof. 

Fix $f\in L^{p(·)}$. By Proposition 3,

$$\rho \left(f\right)=inf\left\{c+I_{A_{\varrho }}(c,X)+I_{A_{\varphi }}(X,f):(c,X)\in \mathbb{R}\times L^{r(·)}(\Omega ,\mathbb{R})\right\}.$$

Minimizing first over $c\in \mathbb{R}$ and then over X with $(X,f)\in A_{\varphi }$, and using the monotonicity B1 of $\varrho$ together with the fact that $\varphi \left(f\right)\in L^{r(·)}(\Omega ,\mathbb{R})$ by A3, we obtain

$$\rho \left(f\right)=inf\left\{\varrho \left(X\right):X\in L^{r(·)}(\Omega ,\mathbb{R}),\varphi \left(f\right)\le X\right\}=\varrho \left(\varphi \left(f\right)\right),$$

the infimum being attained at $X=\varphi \left(f\right)$.

We now dualize the outer functional $\varrho$. Since $\varrho$ is proper, convex and lower semicontinuous on $L^{r(·)}(\Omega ,\mathbb{R})$, the Fenchel–Moreau theorem yields $\varrho \left(X\right)={sup}_{D\in L^{r^{\prime }(·)}(\Omega ,\mathbb{R})}\{\langle D,X\rangle -{\varrho }^{*}\left(D\right)\}$ for every $X\in L^{r(·)}(\Omega ,\mathbb{R})$, where ${\varrho }^{*}\left(D\right)={sup}_{X\in L^{r(·)}(\Omega ,\mathbb{R})}\{\langle D,X\rangle -\varrho \left(X\right)\}$. The monotonicity B1 forces ${\varrho }^{*}\left(D\right)=+\infty$ whenever $D\ngeqq 0$: if $\langle D,X_0\rangle >0$ for some $X_0\le 0$ then B1 gives $\varrho \left(t{X}_0\right)\le \varrho \left(0\right)=0$ for all $t\ge 0$, and hence ${\varrho }^{*}\left(D\right)\ge t\langle D,X_0\rangle \to +\infty$ as $t\to +\infty$. Consequently,

$$\rho \left(f\right)=\underset{D\in L^{r^{\prime }(·)}{(\Omega ,\mathbb{R})}_{+}}{sup}\left\{\langle D,\varphi \left(f\right)\rangle -{\varrho }^{*}\left(D\right)\right\}.$$

(17)

It remains to dualize the pairing $\langle D,\varphi \left(f\right)\rangle$ for fixed $D\in L^{r^{\prime }(·)}{(\Omega ,\mathbb{R})}_{+}$. Since $(Y,f)\in A_{\varphi }$ means $\varphi \left(f\right)\le Y$ and $D\ge 0$, we have $\langle D,Y\rangle \ge \langle D,\varphi \left(f\right)\rangle$ with equality at $Y=\varphi \left(f\right)$, and therefore

$$\langle D,\varphi \left(f\right)\rangle =inf\left\{\langle D,Y\rangle :(Y,g)\in A_{\varphi },g=f\right\}.$$

The right-hand side has the form ${inf}_z\{F\left(z\right)+G(\Lambda z)\}$ on the Banach space $L^{r(·)}(\Omega ,\mathbb{R})\times L^{p(·)}$, where $F(Y,g):=I_{A_{\varphi }}(Y,g)+\langle D,Y\rangle$, $\Lambda (Y,g):=g$ and $G:=I_{\left\{f\right\}}$. Both F and G are proper, convex and lower semicontinuous; the lower semicontinuity of $I_{A_{\varphi }}$ follows from the closedness of $A_{\varphi }$: if $(Y_n,g_n)\to (Y,g)$ in norm with $\varphi \left(g_n\right)\le Y_n$ $\mu$-a.e. then along a subsequence the convergence holds $\mu$-a.e., and the continuity of $\varphi$ gives $\varphi \left(g\right)\le Y$$\mu$-a.e. Moreover, $\Lambda \mathrm{dom}F-\mathrm{dom}G=\{g-f:(Y,g)\in A_{\varphi }\mathrm{for}\mathrm{some}Y\}=L^{p(·)}$, because $(\varphi \left(g\right),g)\in A_{\varphi }$ for every $g\in L^{p(·)}$; in particular, ${\bigcup }_{\lambda >0}\lambda (\Lambda \mathrm{dom}F-\mathrm{dom}G)=L^{p(·)}$ is a closed subspace. Hence the constraint qualification of the Attouch–Brézis duality theorem [31] is satisfied, and we obtain

$$\langle D,\varphi \left(f\right)\rangle =\underset{f^{*}\in {\left(L^{p(·)}\right)}^{*}}{sup}\left\{\langle f^{*},f\rangle -I_{A_{\varphi }}^{*}(-D,f^{*})\right\},$$

(18)

since $F^{*}(0,f^{*})={sup}_{(Y,g)\in A_{\varphi }}\{\langle f^{*},g\rangle -\langle D,Y\rangle \}=I_{A_{\varphi }}^{*}(-D,f^{*})$ and $G^{*}(-f^{*})=-\langle f^{*},f\rangle$.

Combining (17) and (18), and observing that

$${\varrho }^{*}\left(Y^{*}\right)=\underset{X\in L^{r(·)}(\Omega ,\mathbb{R})}{sup}\underset{c\ge \varrho \left(X\right)}{sup}\left\{\langle Y^{*},X\rangle -c\right\}=I_{A_{\varrho }}^{*}(-1,Y^{*}),$$

we arrive at

$$\rho \left(f\right)=\underset{(Y^{*},f^{*})\in L^{r^{\prime }(·)}{(\Omega ,\mathbb{R})}_{+}\times {\left(L^{p(·)}\right)}^{*}}{sup}\left\{\langle f^{*},f\rangle -\alpha (Y^{*},f^{*})\right\},$$

with

$$\alpha (Y^{*},f^{*})=I_{A_{\varrho }}^{*}(-1,Y^{*})+I_{A_{\varphi }}^{*}(-Y^{*},f^{*}).$$

Expanding the two conjugates on the right-hand side gives exactly (16). To conclude, note that extending the supremum from $L^{r^{\prime }(·)}{(\Omega ,\mathbb{R})}_{+}$ to the whole space $L^{r^{\prime }(·)}(\Omega ,\mathbb{R})$ does not change its value. Indeed, if $Y^{*}\ngeqq 0$ then ${\varrho }^{*}\left(Y^{*}\right)=+\infty$ as shown above; and for every $f^{*}$ one also has $I_{A_{\varphi }}^{*}(-Y^{*},f^{*})=+\infty$, since A0 provides $x_0\in E$ with $\varphi \left(x_0\right)=0$, and testing the supremum at $(th,x_0)\in A_{\varphi }$ with $h\in L^{r(·)}{(\Omega ,\mathbb{R})}_{+}$ satisfying $\langle Y^{*},h\rangle <0$ and letting $t\to +\infty$ yields arbitrarily large values. Hence $P=\{(Y^{*},f^{*}):\alpha (Y^{*},f^{*})<\infty \}\subset L^{r^{\prime }(·)}{(\Omega ,\mathbb{R})}_{+}\times L^{p^{\prime }(·)}(\Omega ,E^{*})$, and (15) follows. □

Corollary 1.

Under the assumptions of Theorem 3, the systemic risk measure admits the reduced dual representation

$$\rho \left(f\right)=\underset{D\in L^{r^{\prime }(·)}{(\Omega ,\mathbb{R})}_{+}}{sup}\left\{\langle D,\varphi \left(f\right)\rangle -{\varrho }^{*}\left(D\right)\right\},f\in L^{p(·)},$$

(19)

where ${\varrho }^{*}$ is the Fenchel conjugate of ϱ on $L^{r(·)}(\Omega ,\mathbb{R})$.

Proof. 

This is exactly the identity (17) established in the proof of Theorem 3. □

Remark 14.

Representation (19) clarifies the role of the variable exponent. The dual variable $D\in L^{r^{\prime }(·)}{(\Omega ,\mathbb{R})}_{+}$ plays the role of a scenario density weighting the normalized loss $\varphi \left(f\right)$, while the variable exponent $p(·)$ determines the domain on which the composition $\varrho \circ \varphi$ is defined and, through the conjugate variable $f^{*}\in L^{p^{\prime }(·)}(\Omega ,E^{*})$ in (15), the state-dependent pricing of systemic positions. If the exponent is constant, $p(·)\equiv q$, then $L^{p(·)}(\Omega ,E)=L^q(\Omega ,E)$, and both (15) and (19) reduce to the classical Fenchel–Moreau dual representation of the composition $\varrho \circ \varphi$ on the fixed-exponent space. In this sense, the present framework contains the fixed-exponent theory as a special case.

To derive the dual representation of a coherent systemic risk measure, we again follow the approach used for the systemic risk measure in Theorem 3.

Theorem 4.

Suppose $\rho =\varrho \circ \varphi$, where ϱ is a lower semicontinuous coherent single-firm risk measure and ϕ is a continuous coherent deterministic function. Then, for any $f\in L^{p(·)}$, $\rho \left(f\right)$ has the following form

$$\rho \left(f\right)=\underset{(Y^{*},f^{*})\in {\mathcal{P}}_{\circ }}{sup}\langle f^{*},f\rangle ,$$

(20)

with $\alpha (Y^{*},f^{*})=0$ and ${\mathcal{P}}_{\circ }$ defined by

$${\mathcal{P}}_{\circ }:=\left\{(Y^{*},f^{*})\in L^{r^{\prime }(·)}(\Omega ,\mathbb{R})\times {\left(L^{p(·)}\right)}^{*}\mid (1,Y^{*})\in A_{\varrho }^{\circ },(Y^{*},f^{*})\in A_{\varphi }^{\circ }\right\},$$

where the dual acceptance sets are given by

$$A_{\varrho }^{\circ }:=\left\{(b,X^{*})\in \mathbb{R}\times L^{r^{\prime }(·)}(\Omega ,\mathbb{R})\mid bc-\langle X^{*},X\rangle \ge 0,\forall (c,X)\in A_{\varrho }\right\}$$

and

$$A_{\varphi }^{\circ }:=\left\{(Y^{*},f^{*})\in L^{r^{\prime }(·)}(\Omega ,\mathbb{R})\times {\left(L^{p(·)}\right)}^{*}\mid \langle Y^{*},Y\rangle -\langle f^{*},f\rangle \ge 0,\forall (Y,f)\in A_{\varphi }\right\}.$$

Proof. 

In view of Theorem 3, $\rho \left(f\right)$ admits the representation (15) with penalty $\alpha (Y^{*},f^{*})={\varrho }^{*}\left(Y^{*}\right)+I_{A_{\varphi }}^{*}(-Y^{*},f^{*})$. Since $\varrho$ is positively homogeneous by B4, its conjugate takes only the values 0 and $+\infty$: if $\langle Y^{*},X_0\rangle -\varrho \left(X_0\right)>0$ for some $X_0\in L^{r(·)}(\Omega ,\mathbb{R})$, then replacing $X_0$ by $t{X}_0$ and letting $t\to +\infty$ yields ${\varrho }^{*}\left(Y^{*}\right)=+\infty$. Moreover,

$${\varrho }^{*}\left(Y^{*}\right)=0\iff \langle Y^{*},X\rangle \le c\mathrm{for}\mathrm{all}(c,X)\in A_{\varrho }\iff (1,Y^{*})\in A_{\varrho }^{\circ }.$$

Likewise, the positive homogeneity A4 of $\varphi$ implies that $A_{\varphi }$ is a cone, so its support function $I_{A_{\varphi }}^{*}(-Y^{*},f^{*})$ also takes only the values 0 and $+\infty$, and

$$I_{A_{\varphi }}^{*}(-Y^{*},f^{*})=0\iff \langle f^{*},g\rangle \le \langle Y^{*},Y\rangle \mathrm{for}\mathrm{all}(Y,g)\in A_{\varphi }\iff (Y^{*},f^{*})\in A_{\varphi }^{\circ }.$$

Therefore, $\alpha$ takes only the values 0 and $+\infty$, and $\alpha (Y^{*},f^{*})=0$ if and only if $(Y^{*},f^{*})\in P^{\circ }$. Substituting this into (15), the pairs with $\alpha (Y^{*},f^{*})=+\infty$ contribute nothing to the supremum, and (20) follows. □

Remark 15.

Once the duality theory of the variable-exponent Bochner–Lebesgue space $L^{p(·)}$ is coupled with the systemic risk measure developed in the preceding section, the stochastic fluctuations of the exponent $p(·)$ become an instantaneous projection of market sentiment and local volatility. Theorems 3 and 4 deliver dual representations of systemic risk measures on $L^{p(·)}$ that eliminate fixed sensitivity assumptions and offer regulators a rigorous analytical framework. In extreme market episodes or low-frequency, high-loss events, these representations suggest a conceptual foundation for designing state-dependent policy instruments, such as capital surcharges, liquidity buffers, or position limits, whose sensitivity parameters can be linked to observable market volatility indicators. Consequently, this framework significantly enhances the theoretical accuracy and robustness of systemic risk quantification under pronounced market turbulence.

6. Examples

In this section, we apply the previously introduced systemic risk measures to practical scenarios and verify that each example satisfies all the axioms stated in Section 3 and Section 4. To avoid confusion between the outer state space (macroeconomic regimes) and the inner institution-level space, we first clarify the probabilistic setup.

Probabilistic setup. Let $({\Omega }_1,{\mathcal{F}}_1,{\mu }_1)$ be the state space for the systemic risk factor, and let $({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$ be the institution-level probability space. Throughout this section, we take

$${\Omega }_1=\{{\omega }_1,\dots ,{\omega }_m\},m\ge 2,{\mu }_1\left(\left\{{\omega }_j\right\}\right)>0,j=1,\dots ,m,$$

i.e., the systemic factor admits finitely many macroeconomic regimes (in the numerical illustration below, $m=2$ with a good and a bad regime). The variable exponent $p(·):{\Omega }_1\to [1,\infty )$ is generated by a market volatility indicator $V:{\Omega }_1\to [0,\infty )$ through $p\left(\omega \right)=g\left(V\right(\omega \left)\right)$ as in Assumption 1, with a non-decreasing g (e.g., the threshold specification (3) of Remark 3); since ${\Omega }_1$ is finite, Assumption 2 holds automatically with $1<p^{-}\le p^{+}<\infty$. Because ${\Omega }_1$ is finite, all variable-exponent and fixed-exponent spaces on ${\Omega }_1$ coincide as sets,

$$L^{p(·)}\left({\Omega }_1\right)=L^q\left({\Omega }_1\right)\mathrm{for}\mathrm{every}q\in [1,\infty ),$$

(21)

and every ${\mathcal{F}}_1$-measurable function on ${\Omega }_1$ is integrable. In particular, the range condition in A3 (and likewise in C5) can be verified with an arbitrary $q\in [1,\infty )$, which we choose to match the domain of the single-firm risk measure; this makes the relationship between the variable exponent $p(·)$ and the fixed exponent q in A3 and C5 explicit.

In Examples 2–4, we take $E=L^q({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$ with $q\in (1,\infty )$, $K=L_{+}^q\left({\Omega }_2\right)$ and the numeraire $z\equiv 1$. An element $f\in L^{p(·)}({\Omega }_1,E)$ is identified with a jointly measurable kernel $k:{\Omega }_1\times {\Omega }_2\to \mathbb{R}$ via $f\left(\omega \right)\left(\xi \right)=k(\omega ,\xi )$, $\omega \in {\Omega }_1$, $\xi \in {\Omega }_2$. In Example 5, we instead take $E={\mathbb{R}}^N$ and $K={\mathbb{R}}_{+}^N$.

Remark 16.

The choice $q\in (1,\infty )$ guarantees that E is reflexive and that $E^{*}=L^{q/(q-1)}\left({\Omega }_2\right)$ has the Radon–Nikodým property, as required in Section 2. Note that for $q<\infty$ the cone $K=L_{+}^q\left({\Omega }_2\right)$ has an empty interior, so $z\equiv 1$ is only a quasi-interior point of K (i.e., $\langle y,z\rangle >0$ for every $y\in K_0\setminus \left\{0\right\}$). The interior-point property of z is used only in the proof of Lemma 3.1, which provides a merely sufficient condition for A3; in all the examples below, A3 is verified directly, and A5 and C7 require only the designated element z. Hence the examples remain within the framework. Alternatively, one may relax $z\in int\left(K\right)$ to quasi-interiority in Section 2 without affecting any subsequent result, or take $E={\mathbb{R}}^N$ as in Example 5, where $z=(1,\dots ,1)\in int\left(K\right)$ holds literally.

Example 2.

Let $E=L^q({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$. We define the deterministic function by the Bochner integral

$$\varphi \left(f\right)\left(\omega \right)={\int }_{{\Omega }_2}k(\omega ,\xi )d{\mu }_2\left(\xi \right),\omega \in {\Omega }_1,$$

(22)

i.e., $\varphi :E\to \mathbb{R}$ is given by $\varphi \left(x\right)={\int }_{{\Omega }_2}x\left(\xi \right)d{\mu }_2\left(\xi \right)$, $x\in E$. We verify A0–A3 of Definition 4.

A0 (surjectivity): For any $c\in \mathbb{R}$, $\varphi \left(cz\right)=c{\int }_{{\Omega }_2}d{\mu }_2=c$; hence $\varphi \left(E\right)=\mathbb{R}$, i.e., $\mathfrak{L}=\mathbb{R}$.

A1 (monotonicity): If $x{\ge }_{K}y$, i.e., $x\ge y$ ${\mu }_2$-a.e., then $\varphi \left(x\right)\ge \varphi \left(y\right)$ by the monotonicity of the integral.

A2 (convexity): ϕ is linear, hence convex.

A3 (integrability and range). For every $f\in L^{p(·)}({\Omega }_1,E)$, Hölder’s inequality (recall that ${\mu }_2$ is a probability measure) gives the pointwise estimate

$$\left|\varphi \left(f\left(\omega \right)\right)\right|=\left|{\int }_{{\Omega }_2}k(\omega ,\xi )d{\mu }_2\left(\xi \right)\right|\le \parallel f\left(\omega \right)\parallel ,\omega \in {\Omega }_1,$$

(23)

and $\omega \mapsto \parallel f\left(\omega \right)\parallel$ belongs to $L^{p(·)}\left({\Omega }_1\right)$ by the definition of the Bochner space; no additional kernel condition is needed. Conversely, for any $X\in L^q\left({\Omega }_1\right)$, the constant-section lifting $f\left(\omega \right):=X\left(\omega \right)z$ satisfies $\parallel f\left(\omega \right)\parallel =\left|X\right(\omega \left)\right|$ and $\varphi \left(f\right)=X$. In view of (21), we conclude

$$\left\{f_{\varphi }:{\Omega }_1\ni \omega \mapsto \varphi \left(f\left(\omega \right)\right)\in \mathbb{R}|f\in L^{p(·)}({\Omega }_1,E)\right\}=L^q\left({\Omega }_1\right)\mathit{for}\mathit{every}q\in [1,\infty ).$$

Consider the distortion entropic risk measure ${\varrho }_d$ introduced by Tsanakas and Desli [32], i.e.,

$${\varrho }_d\left(X\right)={\displaystyle \frac{1}{d}}log{Q}_h\left(e^{dX}\right),d>0,$$

(24)

where the distorted expectation $Q_h$ is defined by

$$Q_h\left(X\right):={\int }_{-\infty }^0\left(h\left({\mu }_1(X>x)\right)-1\right)dx+{\int }_0^{\infty }h\left({\mu }_1(X>x)\right)dx,$$

with $h:[0,1]\to [0,1]$ being a non-decreasing, concave function with $h\left(0\right)=0$ and $h\left(1\right)=1$. The concavity of h guarantees that $Q_h$ is convex (submodular), which in turn yields the convexity of ${\varrho }_d$; see [32]. We verify B1–B3 of Definition 5 on $L^q\left({\Omega }_1\right)$ (any $q\ge 1$; since ${\Omega }_1$ is finite, $e^{dX}$ is bounded and ${\varrho }_d$ is finitely valued everywhere).

B1 (monotonicity). $X\ge Y$ a.s. implies $e^{dX}\ge e^{dY}$ a.s.; $Q_h$ is monotone since h is non-decreasing, and log is increasing.

B2 (convexity). For concave h, the distorted expectation $Q_h$ is convex; the convexity of ${\varrho }_d$ then follows from the convexity and monotonicity of the entropic transform [32].

B3 (constancy). For a constant $a\in \mathbb{R}$, $Q_h\left(a\right)=a$ because $h\left(0\right)=0$ and $h\left(1\right)=1$; hence ${\varrho }_d\left(a\right)={\displaystyle \frac{1}{d}}log{e}^{da}=a$.

Hence, we can use the deterministic function (22) and the distortion entropic risk measure to define the distortion entropic systemic risk measure by

$${\rho }_{h,d}\left(f\right)={\displaystyle \frac{1}{d}}log{Q}_h\left(exp\left(d{\int }_{{\Omega }_2}k(·,\xi )d{\mu }_2\left(\xi \right)\right)\right),d>0.$$

By Theorem 1, ${\rho }_{h,d}={\varrho }_d\circ \varphi$ is a convex systemic risk measure on $L^{p(·)}({\Omega }_1,E)$, i.e., it satisfies C0–C5 of Definition 6.

Example 3.

Let $E=L^q({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$. We define the deterministic function by the Bochner integral

$$\varphi \left(f\right)\left(\omega \right)={\int }_{{\Omega }_2}k(\omega ,\xi )d{\mu }_2\left(\xi \right),\omega \in {\Omega }_1.$$

(25)

The verification of A0–A3 is identical to that in Example 2.

We consider the expected shortfall ${\varrho }_e$ discussed by Acharya et al. [8], i.e.

$${\varrho }_e\left(X\right)=\underset{e}{\mathrm{ES}}\left(X\right)=\underset{\mathbb{P}\in {\mathcal{H}}_e}{sup}{\mathbb{E}}_{\mathbb{P}}\left(X\right),$$

(26)

where

$${\mathcal{H}}_e=\left\{\mathbb{P}\ll {\mu }_1|{\displaystyle \frac{d\mathbb{P}}{d{\mu }_1}}\le {\displaystyle \frac{1}{1-e}}\right\},e\in (0,1).$$

We verify B1–B3 of Definition 5 on $L^q\left({\Omega }_1\right)$ (any $q\ge 1$).

B1: ${\mathbb{E}}_{\mathbb{P}}$ is monotone for every $\mathbb{P}\in {\mathcal{H}}_e$, and the supremum preserves the order.

B2: ${\varrho }_e$ is the pointwise supremum of linear (hence convex) functionals, and it is therefore convex.

B3: Every $\mathbb{P}\in {\mathcal{H}}_e$ is a probability measure, so ${\varrho }_e\left(a\right)={sup}_{\mathbb{P}\in {\mathcal{H}}_e}{\mathbb{E}}_{\mathbb{P}}\left(a\right)=a$ for any $a\in \mathbb{R}$.

Moreover, $|{\varrho }_e{\left(X\right)|\le (1-e)}^{-1}{\mathbb{E}}_{{\mu }_1}\left|X\right|$, so ${\varrho }_e$ is finitely valued on $L^q\left({\Omega }_1\right)$. Hence, we can use the deterministic function (25) and the expected shortfall (26) to define the expected shortfall systemic risk measure by

$${\rho }_e\left(f\right)=\underset{\mathbb{P}\in {\mathcal{H}}_e}{sup}{\mathbb{E}}_{\mathbb{P}}\left({\int }_{{\Omega }_2}k(·,\xi )d{\mu }_2\left(\xi \right)\right).$$

By Theorem 1, ${\rho }_e$ is a convex systemic risk measure. Since ϕ in (25) is positively homogeneous with $\varphi \left(z\right)=1$ and $\underset{e}{\mathrm{ES}}$ is coherent, ${\rho }_e$ is in fact a coherent systemic risk measure by Theorem 2.

Example 4.

Let $E=L^q({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$. We define the deterministic function by

$$\varphi \left(f\right)\left(\omega \right)={\left({\int }_{{\Omega }_2}k(\omega ,\xi )d{\mu }_2\left(\xi \right)\right)}^{+},\omega \in {\Omega }_1,$$

(27)

where ${\left(X\right)}^{+}:=max\{X,0\}$. We verify A0–A3 of Definition 4.

A0: $\varphi \left(cz\right)=c^{+}$ for $c\in \mathbb{R}$, so $\varphi \left(E\right)={\mathbb{R}}_{+}$, i.e., $\mathfrak{L}={\mathbb{R}}_{+}$.

A1: $x{\ge }_{K}y$ implies ${\int }_{{\Omega }_2}xd{\mu }_2\ge {\int }_{{\Omega }_2}yd{\mu }_2$, and ${(·)}^{+}$ is non-decreasing; hence $\varphi \left(x\right)\ge \varphi \left(y\right)$.

A2: ϕ is the composition of the linear map $x\mapsto {\int }_{{\Omega }_2}xd{\mu }_2$ with the non-decreasing convex function ${(·)}^{+}$, and it is therefore convex.

A3: The pointwise estimate $\left|\varphi \left(f\left(\omega \right)\right)\right|\le \left|{\int }_{{\Omega }_2}k(\omega ,\xi )d{\mu }_2\left(\xi \right)\right|\le \parallel f\left(\omega \right)\parallel$ holds for every $f\in L^{p(·)}({\Omega }_1,E)$ as in (23). Conversely, for any $X\in L_{+}^q\left({\Omega }_1\right)$, the lifting $f\left(\omega \right)=X\left(\omega \right)z$ gives $\varphi \left(f\right)=X^{+}=X$. In view of (21),

$$\left\{f_{\varphi }:{\Omega }_1\ni \omega \mapsto \varphi \left(f\left(\omega \right)\right)\in {\mathbb{R}}_{+}|f\in L^{p(·)}({\Omega }_1,E)\right\}=L_{+}^q\left({\Omega }_1\right)\mathit{for}\mathit{every}q\in [1,\infty ),$$

so that A3 holds with $\mathfrak{L}={\mathbb{R}}_{+}$.

The loss-based risk measures were first investigated by Cont et al. [33]. Next, we consider the following single-firm risk measure,

$${\varrho }_l\left(X\right)={\mathbb{E}}_{\mathbb{Q}}\left(X\right),$$

(28)

where $\mathbb{Q}$ is a given probability measure on $({\Omega }_1,{\mathcal{F}}_1)$ with $\mathbb{Q}\ll {\mu }_1$ and bounded Radon–Nikodým derivative $d\mathbb{Q}/d{\mu }_1\in L^{\infty }\left({\Omega }_1\right)$. This guarantees that the expectation ${\mathbb{E}}_{\mathbb{Q}}\left[X\right]$ is well defined for all $X\in L^q\left({\Omega }_1\right)$, since ${\mathbb{E}}_{\mathbb{Q}}\left|X\right|\le \parallel d\mathbb{Q}/d{\mu }_1{\parallel }_{\infty }{\mathbb{E}}_{{\mu }_1}\left|X\right|$. B1–B3 follow from the monotonicity and linearity of ${\mathbb{E}}_{\mathbb{Q}}$ together with ${\mathbb{E}}_{\mathbb{Q}}\left(a\right)=a$ for any $a\in \mathbb{R}$. Hence, we can use the deterministic function (27) and the single-firm risk measure (28) to define the loss-based systemic risk measure by

$${\rho }_l\left(f\right)={\mathbb{E}}_{\mathbb{Q}}\left[{\left({\int }_{{\Omega }_2}k(·,\xi )d{\mu }_2\left(\xi \right)\right)}^{+}\right].$$

Since the deterministic function defined above satisfies the positive-range branch $\mathfrak{L}={\mathbb{R}}_{+}$, while ${\varrho }_l$ is defined on the whole space $L^q\left({\Omega }_1\right)$, Proposition 1 shows that ${\rho }_l$ is a convex systemic risk measure with the positive-range branch $\mathfrak{L}={\mathbb{R}}_{+}$.

Example 5

(Nonlinear contagion aggregator). In Examples 2–4, the Bochner integral is a linear aggregator. We now provide a nonlinear aggregator that exhibits such contagion. Let $E={\mathbb{R}}^N$ (with $N\ge 2$ institutions), equipped with the norm $\parallel x\parallel ={\sum }_{i=1}^N\left|x_i\right|$ and the partial order induced by the cone $K={\mathbb{R}}_{+}^N$, and with numeraire $z=(1,\dots ,1)$. Define the deterministic function $\varphi :{\mathbb{R}}^N\to \mathbb{R}$ by

$$\varphi (x_1,\dots ,x_N)=\underset{1\le i\le N}{max}{x}_i+\lambda {\displaystyle \sum _{i=1}^N}{x}_i^{+},\lambda \ge 0,$$

(29)

where $x_i^{+}=max\{x_i,0\}$. We verify A0–A3 of Definition 4.

A0 (surjectivity): For $t\in \mathbb{R}$, $\varphi \left(tz\right)=t+\lambda N{t}^{+}$, which equals t for $t\le 0$ and $(1+\lambda N)t$ for $t\ge 0$; hence $\varphi \left({\mathbb{R}}^N\right)=\mathbb{R}$ and $\mathfrak{L}=\mathbb{R}$.

A1 (monotonicity): The maps $x\mapsto {max}_{1\le i\le N}{x}_i$ and $x\mapsto x_i^{+}$ are non-decreasing in each component; hence ϕ is non-decreasing with respect to ${\ge }_K$.

A2 (convexity): $x\mapsto {max}_{1\le i\le N}{x}_i$ is convex as the maximum of linear functions, each $x\mapsto x_i^{+}$ is convex, and $\lambda \ge 0$; hence ϕ is convex.

A3 (integrability and range): For every $x\in {\mathbb{R}}^N$,

$$|\varphi \left(x\right)|\le \underset{1\le i\le N}{max}|x_i|+\lambda {\displaystyle \sum _{i=1}^N}|x_i|\le (1+\lambda ){\displaystyle \sum _{i=1}^N}|x_i|=(1+\lambda )\parallel x\parallel ,$$

(30)

and this linear bound holds uniformly on ${\mathbb{R}}^N$. Consequently, $|\varphi (f\left(\omega \right))|\le (1+\lambda )\parallel f\left(\omega \right)\parallel$ for every $f\in L^{p(·)}({\Omega }_1,{\mathbb{R}}^N)$, with $\omega \mapsto \parallel f\left(\omega \right)\parallel$ in $L^{p(·)}\left({\Omega }_1\right)$. Conversely, for $X\in L^q\left({\Omega }_1\right)$, define $f\left(\omega \right)=t\left(X\right(\omega \left)\right)z$, where $t\left(u\right)=u$ for $u<0$ and $t\left(u\right)=u/(1+\lambda N)$ for $u\ge 0$; then $\left|t\right(u\left)\right|\le \left|u\right|$, $f\in L^{p(·)}({\Omega }_1,{\mathbb{R}}^N)$ and $\varphi \left(f\right)=X$ by the computation in A0. In view of (21), the range set $\{f_{\varphi }:{\Omega }_1\ni \omega \mapsto \varphi \left(f\left(\omega \right)\right)\in \mathbb{R}|f\in L^{p(·)}({\Omega }_1,{\mathbb{R}}^N)\}$ equals $L^q\left({\Omega }_1\right)$ for every $q\in [1,\infty )$.

Remark 17.

The specification (29) corresponds to the case $\alpha =1$ of the more general candidate ${max}_{1\le i\le N}{x}_i+\lambda {\left({\sum }_{i=1}^N{x}_i^{+}\right)}^{\alpha }$, $\alpha \in (0,1]$. For $\alpha <1$, this candidate does not satisfy A2: e.g., for $N=1$, $\lambda =1$, $\alpha =1/2$, the function $\psi \left(x\right)=x+\sqrt{x^{+}}$ satisfies $\psi \left(1.5\right)\approx 2.725>\frac{1}{2}\left(\psi \left(1\right)+\psi \left(2\right)\right)\approx 2.707$. Moreover, $t^{\alpha }>t$ for $t\in (0,1)$, so a linear bound of the type ${\left({\sum }_i{x}_i^{+}\right)}^{\alpha }\le {\sum }_i\left|x_i\right|$ fails uniformly near zero, and the integrability argument in A3 breaks down. A genuinely nonlinear-power alternative that preserves convexity is

$${\varphi }_r\left(x\right)=\underset{1\le i\le N}{max}{x}_i+\lambda {\left({\displaystyle \sum _{i=1}^N}{\left(x_i^{+}\right)}^r\right)}^{1/r},r\in [1,\infty ),$$

which is convex (the ${\ell }^r$-norm is convex and non-decreasing on ${\mathbb{R}}_{+}^N$, and $x\mapsto x^{+}$ is convex) and satisfies ${\left({\sum }_i{\left(x_i^{+}\right)}^r\right)}^{1/r}\le {\sum }_i\left|x_i\right|$; the verification of A0–A3 for ${\varphi }_r$ is analogous.

Combining (29) with the expected shortfall ${\varrho }_e$ from Example 3, we obtain the nonlinear contagion systemic risk measure

$${\rho }_{cont}\left(f\right)=\underset{e}{\mathrm{ES}}\left(\underset{1\le i\le N}{max}{f}_i(·)+\lambda {\displaystyle \sum _{i=1}^N}{f}_i^{+}(·)\right).$$

Unlike the linear Bochner integral, this measure satisfies $\varphi (x+y)\ne \varphi \left(x\right)+\varphi \left(y\right)$ in general: e.g., for $N=2$, $\varphi \left(\right(1,0\left)\right)+\varphi \left(\right(0,1\left)\right)=2+2\lambda \ne 1+2\lambda =\varphi \left(\right(1,1\left)\right)$. It thereby reflects the non-additive societal impact of heterogeneous equity ownership structures discussed in Chen et al. [1].

Remark 18.

The function ϕ in (29) is positively homogeneous (A4), and $\tilde{\varphi }:=\varphi /(1+\lambda N)$ satisfies the normalization $\tilde{\varphi }\left(z\right)=1$ (A5). Since $\underset{e}{\mathrm{ES}}$ is coherent (B4–B5), ${\tilde{\rho }}_{cont}:={\varrho }_e\circ \tilde{\varphi }$ is a coherent systemic risk measure by Theorem 2.

Remark 19.

All deterministic functions ϕ in Examples 2–5 are continuous (in fact Lipschitz) on E, and all single-firm risk measures ϱ are continuous on $L^q\left({\Omega }_1\right)$, since ${\Omega }_1$ is finite. In particular, ϱ is lower semicontinuous, so the dual representations of Theorem 3 (and of Theorem 4 in the coherent cases) apply to every example in this section.

We provide a minimal numerical illustration that highlights how the variable exponent $p(·)$ dynamically adjusts the sensitivity of the systemic risk framework to market volatility, in contrast to a conventional fixed-exponent benchmark. Consider the outer state space ${\Omega }_1=\{{\omega }_g,{\omega }_b\}$ with ${\mu }_1\left({\omega }_g\right)={\mu }_1\left({\omega }_b\right)=0.5$, representing a good (low-volatility) and a bad (high-volatility) macroeconomic regime. The variable exponent is

$$p\left({\omega }_g\right)=2,p\left({\omega }_b\right)=4,$$

which is generated, for instance, by the threshold mapping (3) applied to a volatility indicator V with $V\left({\omega }_g\right)\le v^{*}<V\left({\omega }_b\right)$. Thus, the risk measure applies a milder quadratic penalty in the good state and a more severe quartic penalty in the bad state, where tail risk is pronounced.

Suppose the aggregate systemic risk position (already processed by a deterministic function $\varphi$) yields the scalar random variable $Z\in L^{p(·)}\left({\Omega }_1\right)$ with

$$Z\left({\omega }_g\right)=4,Z\left({\omega }_b\right)=10.$$

The modular ${\rho }_{p(·)}\left(Z\right)={\mathbb{E}}_{{\mu }_1}{\left[\right|Z|}^{p(·)}]$ serves as a natural measure of risk-adjusted penalization intensity. We compare three specifications in

Table 1. Comparison of risk-adjusted modulars under fixed and variable exponents.

Model	Exponent	Modular $\mathit{\rho }\left(\mathit{Z}\right)$	Value
Fixed $q=2$	$q=2$ everywhere	$0.5\times 4^2+0.5\times {10}^2$	58
Fixed $q=4$	$q=4$ everywhere	$0.5\times 4^4+0.5\times {10}^4$	5128
Variable $p(·)$	$p\left({\omega }_g\right)=2$, $p\left({\omega }_b\right)=4$	$0.5\times 4^2+0.5\times {10}^4$	5008

.

The fixed $q=2$ model severely underestimates the risk contribution of the bad state (penalizing it only quadratically), yielding a total modular of 58. The fixed $q=4$ model, by contrast, over-penalizes the good state (raising $4^2=16$ to $4^4=256$), producing a modular of 5128. The variable-exponent modular 5008 is almost identical to the bad-state contribution under $q=4$ (which is 5000), yet it is far more lenient in the good state (8 vs. 128). This demonstrates that $p(·)$ acts as an endogenous risk-sensitivity adjuster: it amplifies tail sensitivity exactly where volatility is high, without distorting the measurement in tranquil periods.

Although the systemic risk measure $\rho$ need not coincide with the Luxemburg norm, the latter governs the geometry of the acceptance set and therefore serves as a natural proxy for the relative severity of risk assessment across specifications. The Luxemburg norm ${\parallel Z\parallel }_{L^{p(·)}}=inf\{\lambda >0:{\rho }_{p(·)}(Z/\lambda )\le 1\}$ yields

$${\parallel Z\parallel }_{L^2}=\sqrt{58}\approx {7.62,\parallel Z\parallel }_{L^4}={5128}^{1/4}\approx 8.46,{\parallel Z\parallel }_{L^{p(·)}}\approx 8.65.$$

Notably, the variable-exponent norm exceeds the fixed $q=4$ norm by roughly $2\%$. This occurs because the Luxemburg norm must equilibrate the state-dependent penalization: the lower exponent $p\left({\omega }_g\right)=2$ prevents the good-state contribution from being compressed to near zero (as it would under the quartic fixed exponent), thereby raising the overall scaling factor $\lambda$. Nevertheless, the variable-exponent norm remains far closer to the high-volatility benchmark than to the low-volatility one ($8.65$ vs. $7.62$), confirming that the systemic risk framework effectively inherits the heightened sensitivity of the bad state.

A regulator relying on the fixed $q=2$ framework would perceive the risk magnitude as roughly $12\%$ lower than the variable-exponent measure ($7.62$ vs. $8.65$). In a volatile environment where the bad state is realized, this underestimation could lead to insufficient capital buffers. The variable-exponent framework eliminates this bias by construction.

Finally,

Table 2. Axiom verification summary for the examples. Here ✓ indicates that the axiom is verified in the corresponding example.

Example	Deterministic Function $\mathit{\varphi }$				Single-Firm Measure $\mathit{\varrho }$			Conclusions
	A0	A1	A2	A3	B1	B2	B3
Ex. 1 (${\rho }_{h,d}$)	✓	✓	✓	✓	✓	✓ (h concave)	✓	convex, Theorem 1
Ex. 2 (${\rho }_e$)	✓	✓	✓	✓	✓	✓	✓	convex/coherent, Theorems 1 and 2
Ex. 3 (${\rho }_l$)	✓	✓	✓	✓	✓	✓	✓	convex, Theorem 1
Ex. 4 (${\rho }_{cont}$)	✓	✓	✓	✓	✓	✓	✓	convex/coherent, Theorems 1 and 2

summarizes the axiom verification carried out in Examples 2–5.

7. Conclusions

Systemic risk poses the greatest threat to financial stability, and its accurate quantification is essential for economic security and public trust. However, existing systemic risk measures are not adequately adapted to market volatility.

In this paper, we adopt the variable-exponent Bochner–Lebesgue space $L^{p(·)}$ to model systemic risk positions under market volatility, where the random exponent $p(·)$ is used to characterize the fluctuations in the financial market. Our principal contribution is the development of systemic risk measures on $L^{p(·)}$ together with their dual representations.

Our framework is built upon two functions. First, a deterministic function dynamically normalizes systemic risk positions, converting multi-factor systemic risk into a measurable variable driven by a single factor, thereby filtering out market volatility noise. Second, a single-firm risk measure is introduced to evaluate the resulting univariate risk exposure. Next, by leveraging the duality of the space $L^{p(·)}$, we derive a complete dual representation of the proposed systemic risk measures. Moreover, in the final section, we provide several concrete examples of the resulting systemic risk measures.

For potential future research, a natural direction is to apply systemic risk measures on $L^{p(·)}$ to some specific risk quantification problems. Another promising direction is to model the drivers of the variable exponent $p(·)$ through information structures and behavioral market states. For instance, the strategic competition among informed insiders and overconfident market makers analyzed by Daher and Damrah [16] provides a concrete game-theoretic micro-foundation for linking market-microstructure noise to the time-varying sensitivity encoded by $p(·)$. Embedding such equilibrium models directly into the construction of the volatility-to-exponent mapping g could yield empirically testable restrictions and sharpen the design of state-dependent regulatory instruments such as capital surcharges and liquidity buffers.