Dynamic Set-Valued Risk Measures

• Dynamic set-valued risk measures are mathematical tools that quantify multi-asset, time-evolving capital requirements while accounting for market frictions.
• They leverage recursive structures, dual representations, and backward stochastic equations to provide robust risk assessments in multicurrency and transaction cost models.
• Applications include systemic risk management, superhedging under transaction costs, and numerical algorithms for vector optimization in financial contexts.

A dynamic set-valued risk measure is a mathematical object used to quantify the multi-asset, time-evolving capital requirements for contingent positions in stochastic financial models. Unlike scalar risk measures, which yield single-valued capital requirements, set-valued risk measures return capital requirement sets—typically in eligible portfolio spaces—thus capturing multidimensionality, trading constraints, and market frictions. The dynamic aspect refers to the measurement at multiple time points, consistent with a given filtration, and adapted to information evolution; set-valued risk measures naturally appear in models with transaction costs, multicurrency requirements, or systemic risk management. The rigorous properties, dual representations, and recursive structures of dynamic set-valued risk measures have driven substantial literature, with central results characterizing time consistency, multiportfolio time consistency (MPTC), and supermartingale properties (Feinstein et al., 2015, Feinstein et al., 2012, Ararat et al., 2019, Chen et al., 2021, Feinstein et al., 2015, Ararat et al., 2020).

1. Formal Definition and Mathematical Framework

On a filtered probability space $$(\Omega, \mathcal F, (\mathcal F_t)_{t=0}^T, \mathbb P)$$, let $L^p(\mathbb R^d)$ denote the space of $\mathbb R^d$-valued, $\mathcal F_T$-measurable random vectors with finite $p$-norm, and $M_t \subseteq L^p_t(\mathbb R^d)$ a closed linear subspace of “eligible portfolios.” A (normalized) time-$t$ set-valued risk measure is a map

$R_t: L^p(\mathbb R^d) \to \mathcal P(M_t)$

satisfying:

• Translativity: $R_t(X + m_t) = R_t(X) - m_t$ for all $m_t \in M_t$;
• Monotonicity: $Y \ge X$ a.s. $\implies R_t(Y) \supseteq R_t(X)$;
• Finiteness at zero: $\emptyset \subsetneq R_t(0) \subsetneq M_t$;
• Normalization: $R_t(X) = R_t(X) + R_t(0)$.

Acceptance sets $A_t = \{ X \in L^p(\mathbb R^d): 0 \in R_t(X) \}$ provide an equivalent description: $R_t(X) = \{ u \in M_t : X + u \in A_t \}$. Random closed sets, selectors, and upper sets (i.e., sets closed under addition by the eligible cone $M_{t,+}$) are canonical in this formalism (Feinstein et al., 2015, Feinstein et al., 2012).

2. Multiportfolio Time Consistency and Recursion

Time consistency for set-valued risk measures admits multiple generalizations. The strongest is multiportfolio time consistency (MPTC), defined by

$$R_s(X) \subseteq \bigcup_{Y \in \mathcal Y} R_s(Y) \implies R_t(X) \subseteq \bigcup_{Y \in \mathcal Y} R_t(Y)$$

for all $0 \le t < s \le T$, all $X$, and all families $\mathcal Y$. MPTC is strictly stronger than scalar time consistency and in multi-dimensional settings is equivalent to a recursive, Bellman-like structure:

$R_t(X) = \bigcup_{Z \in R_s(X)} R_t(-Z)$

for $t < s$. This recursive property is also equivalent to additivity of acceptance sets: $A_t = A_s + (A_t \cap M_s)$ (Feinstein et al., 2015, Feinstein et al., 2012). In scalar ($d=1$) cases, MPTC and classic time consistency coincide; in higher dimension, many natural measures (e.g., AV@R) may fail MPTC unless specifically adapted (Feinstein et al., 2012).

3. Dual Representations and Supermartingale Properties

Duality and scalarization are central to both the analysis and computation of set-valued risk measures. Any set-valued, convex, and closed $R_t$ can be represented, via dual variables $(Q, w)$ (probability measures and nonnegative weight processes), as

$$R_t(X) = \bigcap_{(Q,w) \in W_t} \left( \mathbb{E}_t^Q[-X] + G_t(w) - \alpha_t(Q, w) \right)$$

where $G_t(w)$ is a supporting halfspace and $\alpha_t(Q, w)$ is the (minimal) penalty function (Feinstein et al., 2015, Chen et al., 2021). For any $w \in L^q_t(\mathbb R^d_+)$, scalar risk measures

$\rho_t^w(X) = \essinf_{u \in R_t(X)} w^\mathsf{T} u$

admit dual representations involving conditional expectations and worst-case scenarios.

A pivotal result is the supermartingale relation for multiportfolio time-consistent risk measures:

$$V_t^{(Q, w)}(X) \subseteq \mathbb{E}_t^Q\left[ V_s^{(Q, w_t^s(Q, w))}(X) \right], \quad 0 \le t < s \le T,$$

where $V_t^{(Q, w)}(X)$ is the closed Minkowski sum $R_t(X) + \beta_t(Q, w)$; equality (martingale property) holds for extremal or worst-case dual variables that attain the dual representation (Feinstein et al., 2015).

4. Backward Stochastic Equations and Stochastic Dynamic Programming

Set-valued dynamic risk measures can be described by operator-valued difference equations or inclusions, paralleling the BSDE theory for scalar measures (Ararat et al., 2019, Ararat et al., 2020). In discrete time, the two main forms are:

• Backward stochastic difference inclusions (BSDI), which recursively characterize selectors in acceptance sets. For $Y(t_k) \in L^2(\mathcal{F}_{t_k})$,

$Y(t_{k-1}) \in Y(t_k) + G(t_{k-1}, \psi(t_{k-1}))$

where $G$ is a set-valued driver constructed from the one-step risk measure and random noise increments.

• Set-valued backward stochastic equations (SV-BSAE/BSDE), evolving the acceptance sets as entire objects via dynamic programming recursions that intersect supporting halfspaces corresponding to admissible weights.

Continuous-time analogues involve set-valued BSDEs using specialized operations such as Minkowski addition and the Hukuhara difference for closed convex sets. Well-posedness (existence/uniqueness) of such equations can be established under uniform Lipschitz conditions for the driver and square-integrability of data (Ararat et al., 2020).

5. Examples and Applications

Key examples of dynamic set-valued risk measures demonstrating time consistency and supermartingale properties include:

• Convex entropic risk: Incorporates exponential penalty and yields a supermartingale in terms of conditional relative entropies.
• Dynamic composition of AV@R (Average Value at Risk): The usual multivariate AV@R is not MPTC, but dynamic composition (e.g., via backward recursion) restores the property.
• Superhedging under transaction costs: The acceptance set comprises all positions that can be super-hedged via admissible trading strategies. The resulting risk measure is coherent, closed, and MPTC, naturally induced by market cones and enabling linear vector optimization at each node in a scenario tree (Feinstein et al., 2015).
• Aggregation-based systemic risk measures: Capital requirements are determined via a risk-aggregator function $\Lambda:\mathbb{R}^d \to \mathbb{R}$. Provided $\Lambda$ and the underlying scalar measure are time-consistent, the induced multivariate measure is MPTC and admits a supermartingale representation.

These models commonly translate to backward induction algorithms for computation and have direct interpretations in multi-asset markets, systemic risk, and capital adequacy computation (Feinstein et al., 2015, Feinstein et al., 2012, Feinstein et al., 2015).

6. Algorithmic and Numerical Aspects

Set-valued Bellman’s principle provides a node-wise backward-recursive algorithm for dynamic risk computation, especially tractable on finite event trees and under polyhedrality or convexity assumptions (Feinstein et al., 2015). At each node of the tree, one solves a vector optimization problem—linear in the polyhedral case, convex in the entropic or relaxed worst-case settings—which yields the upper image representing capital requirement sets. The complexity scales with the number of nodes and constraints per node; in the polyhedral case, Benson’s algorithm enables practical computation.

Forward extraction of hedging or capital-injection policies (Bellman-optimal strategies) is possible by barycentric combinations of computed upper image points along realized scenario paths. In high-dimensional and continuous-time settings, numerical solution of set-valued BSDEs or SV-BSAEs remains an active area of research due to the challenges of measurable selection and set integration (Feinstein et al., 2015, Ararat et al., 2020, Ararat et al., 2019).

7. Generalizations, Implications, and Open Problems

Set-valued risk measures unify scalar, vector, and process-based risk measurement frameworks. The theory accommodates markets with frictions, systemic risk, and dynamically updated requirements. Duality and the supermartingale characterization furnish both theoretical tractability and numerically implementable algorithms; the recognition of MPTC as the correct multi-dimensional analogue of time consistency is central for recursive computation (Feinstein et al., 2015, Feinstein et al., 2012).

Current research directions include: extending existence/uniqueness results for set-valued BSDEs in continuous time, refining dual representation theory for process-valued risk measures, and developing efficient numerical schemes for high-dimensional dynamic risk problems (Ararat et al., 2020, Chen et al., 2021, Ararat et al., 2019). A plausible implication is that further advances in set-valued stochastic analysis will yield dynamic programming methodologies applicable beyond financial mathematics, in broader multivariate or networked stochastic optimization domains.