A General Theory of Risk Sharing

Abstract: We introduce a new paradigm for risk sharing that generalizes earlier models based on discrete agents and extends them to allow for sharing risk within a continuum of agents. Agents are represented by points of a measure space and have potentially heterogeneous risk preferences modeled by risk measures. The existence of risk minimizing allocations is proved when constrained to satisfy economically convincing conditions. In the unconstrained case, we derive the dual representation of the value function using a Strassen-type theorem for the weak-star topology. These results are illustrated by explicit formulas when risk preferences are within the family of entropic and expected shortfall risk measures.

• The paper develops a novel mathematical model extending risk sharing to a continuum of agents using measure-theoretic techniques.
• The methodology uses advanced integration methods and dual representation to derive optimal risk allocations under both constrained and unconstrained settings.
• Implications include enhanced financial market regulation and robust modeling of heterogeneous risk preferences in real-world economic systems.

A General Theory of Risk Sharing

Introduction

The paper proposes an advanced theoretical framework for risk sharing that extends classical models involving a finite number of discrete agents to incorporate a continuum of agents. The framework raises important considerations for economic allocations and regulatory practices by addressing the limitations that arise from treating economically significant entities and smaller, diffuse agents as homogenous with regard to their impact on risk models.

Problem Formulation

The risk-sharing problem is mathematically represented as minimizing a total risk function given by:

$\sum_{a\in A}\varrho_{a}(X_{a})\rightarrow\min$

subject to a constraint that ensures the aggregate risk allocation $X_a$ across all agents sums to a predefined total loss $\mathcal{X}$. This setup is generalizable using measure theory, where agents interact across a measure space $A$ with risk preferences characterized by individual risk measures $\varrho_{a}$.

Mathematical Framework

To accommodate a continuum of agents, the framework applies advanced measure-theoretic methods. Agents are represented as measures on a space $(A, \mathscr{A}, \mu)$, permitting both atomic (discrete) and non-atomic (continuum) agent structures. The paper navigates the complexities associated with integrating risk over potentially infinite-dimensional spaces.

Key challenges, such as defining measurable allocations and ensuring integrability under these settings, are resolved by utilizing a topology derived from $L^{\infty}(\mathbb{P})$ and Gelfand integration techniques. The treatment of risk preferences through Cash Additivity, Monotonicity, and the Fatou Properties ensures model reliability and economic soundness.

Constrained Risk Sharing

The framework is extended to accommodate economically realistic constraints—such as those preventing excessive losses or profits for any agent. Constraints are shown to lead to the existence of solutions using techniques that convert the Gelfand integrals to Bochner integrals, leveraging classical results on Lebesgue-Bochner function spaces.

Unconstrained Risk Sharing

For unconstrained settings, the paper delineates conditions under which the integrated risk measure $\operatorname{\Box}_{a\in A}\varrho_{a}\mu(da)$ remains robust (i.e., possesses properties like global finiteness and the Fatou property). A significant result is the dual representation of the value function involving convex conjugates, characterizing the solutions with an integral form over dual spaces formed by individual risk measure representations.

Implications and Examples

The proposed model is applied to specific risk measure families—such as Entropic Risk Measures and Expected Shortfall—showing its broad applicability. A novel notion introduced is the inflation of risk measures, extending the paradigms of calculation to involve variable risk aversion parameters captured through inflation factors. For certain settings, explicit and practical optimal risk allocations are derived, contrasting with certain results derived in the finite-agent literature.

Conclusion

This theoretical advancement has implications for managing risk in financial markets and regulatory settings, particularly under complex scenarios that blend large, impactful agents with myriad smaller ones. The paper's results not only unify several disparate threads from the risk-sharing literature but provide the groundwork for more precise and inclusive economic and policy models incorporating heterogeneous risk preferences.

The research provides strong mathematical underpinnings approved to handle real-world complexities seen in global economic systems, opening numerous avenues for future research in theoretical, regulatory, and practical applications of risk sharing.