Research

This paper develops a comprehensive framework for collective risk measures, tools designed to quantify the aggregate risk stemming from a collective of agents. Crucially, these measures explicitly account for inter-agent cooperation, allowing agents to exchange risk through state-dependent transfers without requiring external capital flows. We review previous works on no-arbitrage in the collective framework, and introduce collective risk measures for both random variables and stochastic processes in discrete time. In the latter case, we allow for time-dependent cooperation and risk sharing, supporting a consistent evaluation of evolving financial positions within a collective framework. Among the several applications of the theory, we study collective super-replication prices and we provide dual characterizations of collective risk measures via families of (martingale) measures.

We introduce the notions of Collective Arbitrage and of Collective Super-replication in a setting where agents are investing in their markets and are allowed to cooperate through exchanges. We accordingly establish versions of the fundamental theorem of asset pricing and of the pricing-hedging duality. Examples show the advantage of our approach.

This paper builds on "Collective Arbitrage and the Value of Cooperation" by Biagini et al. (2025, forthcoming in "Finance and Stochastics"), which introduced in discrete time the notions of collective arbitrage and super-replication in a multi-agent market framework, where agents may operate in several submarkets and collaborate through risk exchange mechanisms.

Expanding on these foundations, we establish a First Fundamental Theorem of Asset Pricing and a collective pricing-hedging duality under different assumptions and with new techniques compared to Biagini et al. (2025). We further introduce the notion of collective replication in order to study collective market completeness and provide a Second Fundamental Theorem of Asset Pricing in this cooperative multi-agent setting.

We consider a family of conditional nonlinear expectations defined on the space of bounded random variables and indexed by the class of all the sub-sigma-algebras of a given underlying sigma-algebra. We show that if this family satisfies a natural consistency property, then it collapses to a conditional certainty equivalent defined in terms of a state-dependent utility function.

This result is obtained by embedding our problem in a decision theoretical framework and providing a new characterization of the Sure-Thing Principle. In particular we prove that this principle characterizes those preference relations which admit consistent backward conditional projections. We build our analysis on state-dependent preferences for a general state space as in Wakker and Zank [Math. Oper. Res., 24 (1999), pp. 8–34] and show that their numerical representation admits a continuous version of the state-dependent utility. In this way, we also answer positively to a conjecture posed in the aforementioned paper.

We extend the framework introduced in "Collective Arbitrage and the Value of Cooperation" by F. Biagini, A. Doldi, J.-P. Fouque, M. Frittelli, and T. Meyer-Brandis (arXiv: 2306.11599v2, 2024) in order to analyze collective dynamic risk measures. In segmented markets, we explore the implications of cooperation on dynamic risk measurement, focusing particularly on aggregation and time consistency.

We consider the problem of optimally sharing a financial position among agents with potentially different reference risk measures. The problem is equivalent to computing the infimal convolution of the risk metrics and finding the so-called optimal allocations. We propose a neural network-based framework to solve the problem and we prove the convergence of the approximated inf-convolution, as well as the approximated optimal allocations, to the corresponding theoretical values.

We support our findings with several numerical experiments.

In this paper, we give an overview of (nonlinear) pricing-hedging duality and of its connection with the theory of entropy martingale optimal transport (EMOT), recently developed, and that of convex risk measures. Similarly to Doldi and Frittelli (2023), we here establish a duality result between a convex optimal transport and a utility maximization problem. Differently from Doldi and Frittelli (2023), we provide here an alternative proof that is based on a compactness assumption.

Subhedging and superhedging can be obtained as applications of the duality discussed above. Furthermore, we provide a dual representation of the generalized Optimized Certainty Equivalent associated to indirect utility.

Shortfall systemic (multivariate) risk measures ρ defined through an N-dimensional multivariate utility function U and random allocations can be represented as classical (one dimensional) shortfall risk measures associated to an explicitly determined 1-dimensional function constructed from U. This finding allows for simplifying the study of several properties of ρ, such as dual representations, law invariance and stability.

In this work, we propose deep learning-based algorithms for the computation of systemic shortfall risk measures defined via multivariate utility functions. We discuss the key related theoretical aspects, with a particular focus on the fairness properties of primal optima and associated risk allocations. The algorithms we provide allow for learning primal optimizers, optima for the dual representation and corresponding fair risk allocations.

We test our algorithms by comparison to a benchmark model, based on a paired exponential utility function, for which we can provide explicit formulas. We also show evidence of convergence in a case in which explicit formulas are not available.

The objective of this paper is to develop a duality between a novel entropy martingale optimal transport (EMOT) problem and an associated optimisation problem. In EMOT, we follow the approach taken in the entropy optimal transport (EOT) problem developed in Liero et al. (Invent. Math. 211:969–1117, 2018), but we add the constraint, typical of martingale optimal transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures.

In the associated problem, the objective functional, related via Fenchel conjugacy to the entropic term in EMOT, is no longer linear as in (martingale) optimal transport. This leads to a novel optimisation problem which also has a clear financial interpretation as a nonlinear subhedging problem. Our theory allows us to establish a nonlinear robust pricing–hedging duality which also covers a wide range of known robust results.

We also focus on Wasserstein-induced penalisations and study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT.

Given a real valued functional T on the space of bounded random variables, we investigate the problem of the existence of a conditional version of nonlinear means. We follow a seminal idea by Chisini (1929), defining a mean as the solution of a functional equation induced by T. We provide sufficient conditions which guarantee the existence of a (unique) solution of a system of infinitely many functional equations, which will provide the so-called Conditional Chisini mean.

We apply our findings in characterizing the scalarization of conditional Risk Measures, an essential tool originally adopted by Detlefsen and Scandolo (2005) to deduce the robust dual representation.

A Systemic Optimal Risk Transfer Equilibrium (SORTE) was introduced in: “Systemic optimal risk transfer equilibrium”, Mathematics and Financial Economics (2021), for the analysis of the equilibrium among financial institutions or in insurance-reinsurance markets. A SORTE conjugates the classical Bühlmann’s notion of a risk exchange equilibrium with a capital allocation principle based on systemic expected utility optimization.

In this paper we extend such a notion to the case when the value function to be optimized is multivariate in a general sense, and it is not simply given by the sum of univariate utility functions. This takes into account the fact that preferences of single agents might depend on the actions of other participants in the game.

Technically, the extension of SORTE to the new setup requires developing a theory for multivariate utility functions and selecting at the same time a suitable framework for the duality theory. Conceptually, this more general framework allows us to introduce and study a Nash Equilibrium property of the optimizer. We prove existence, uniqueness, and the Nash Equilibrium property of the newly defined Multivariate Systemic Optimal Risk Transfer Equilibrium.

We investigate to which extent the relevant features of (static) Systemic Risk Measures can be extended to a conditional setting. After providing a general dual representation result, we analyze in greater detail Conditional Shortfall Systemic Risk Measures. In the particular case of exponential preferences, we provide explicit formulas that also allow us to show a time consistency property.

Finally, we provide an interpretation of the allocations associated to Conditional Shortfall Systemic Risk Measures as suitably defined equilibria. Conceptually, the generalization from static to conditional Systemic Risk Measures can be achieved in a natural way, even though the proofs become more technical than in the unconditional framework.

We describe the axiomatic approach to real-valued Systemic Risk Measures, which is a natural counterpart to the nowadays classical univariate theory initiated by Artzner et al. in the seminal paper “Coherent measures of risk”, Math. Finance, (1999). In particular, we direct our attention towards Systemic Risk Measures of shortfall type with random allocations, which consider as eligible, for securing the system, those positions whose aggregated expected utility is above a given threshold.

We present duality results, which allow us to motivate why this particular risk measurement regime is fair for both the single agents and the whole system at the same time. We relate Systemic Risk Measures of shortfall type to an equilibrium concept, namely a Systemic Optimal Risk Transfer Equilibrium, which conjugates Bühlmann’s Risk Exchange Equilibrium with a capital allocation problem at an initial time. We conclude by presenting extensions to the conditional, dynamic framework.

The latter is the suitable setup when additional information is available at an initial time.

We propose a novel concept of a Systemic Optimal Risk Transfer Equilibrium (SORTE), which is inspired by the Bühlmann’s classical notion of an Equilibrium Risk Exchange. We provide sufficient general assumptions that guarantee existence, uniqueness, and Pareto optimality of such a SORTE. In both the Bühlmann and the SORTE definition, each agent is behaving rationally by maximizing his/her expected utility given a budget constraint. The two approaches differ by the budget constraints.

In Bühlmann’s definition the vector that assigns the budget constraint is given a priori. On the contrary, in the SORTE approach, the vector that assigns the budget constraint is endogenously determined by solving a systemic utility maximization. SORTE gives priority to the systemic aspects of the problem, in order to optimize the overall systemic performance, rather than to individual rationality.

This paper complements and extends Doldi, Frittelli and Maggis, Collective completeness and pricing-hedging duality, Math. Finan. Econ. 19, 757-784 (2025), by studying collective pricing and hedging when admissible risk exchanges form a finitely generated convex cone. The collective First Fundamental Theorem of Asset Pricing and the collective pricing-hedging duality are extended to this setting.

A key contribution is a closedness result showing that no collective arbitrage implies the closedness of the aggregate feasibility cone combining infinite-dimensional trading opportunities with finite-dimensional exchanges. The paper also proves that no-collective-arbitrage prices for vectors of contingent claims form a relatively open convex set. Finally, strong collective replicability is introduced and shown to be equivalent to price uniqueness.

This leads to an enhanced collective Second Fundamental Theorem of Asset Pricing, providing equivalent characterizations of collective completeness and strong collective completeness in terms of the uniqueness of the collective equivalent martingale measure. We highlight that several core aspects of the theory are substantially altered when exchanges belong to a convex cone rather than a vector space.

Within a general semimartingale framework, we study the relationship between collective market efficiency and individual rationality. We derive a necessary and sufficient condition for the existence of (possibly zero-sum) exchanges among agents that strictly increase their indirect utilities and characterize this condition in terms of the compatibility between agents' preferences and collective pricing measures.

The framework applies to both continuous- and discrete-time models and clarifies when cooperation leads to a strict improvement in each participating agent's indirect utility.