A General Theory of Risk Sharing

TL;DR

This paper develops a comprehensive risk sharing framework for a continuum of agents with heterogeneous preferences, providing dual representations and explicit formulas, and explores implications for large market efficiency.

Contribution

It introduces a generalized risk sharing model for a continuum of agents, extending previous discrete models and deriving dual representations and explicit formulas.

Findings

Dual representation of the value function established.

Explicit formulas for entropic and expected shortfall risk measures provided.

Application to Pareto efficiency in large markets demonstrated.

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 on a separable probability space. We derive the dual representation of the value function using a Strassen-type theorem for the weak-star topology and provide a characterization of the acceptance set using Aumann integration. These results are illustrated by explicit formulas when risk preferences are within the family of entropic and expected shortfall risk measures, and applications to Pareto efficiency in large markets.