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A General Theory of Risk Sharing

Vasily Melnikov

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.

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 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.

Paper Structure

This paper contains 26 sections, 20 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Allocations
  4. Risk Preferences
  5. The Risk Sharing Problem
  6. The Unconstrained Value Function
  7. An Integral Infimal Convolution
  8. Dual Representations
  9. Examples
  10. Dilated Risk Measures
  11. Inflated Risk Measures
  12. Applications to Pareto Efficiency
  13. The Weak-Star Strassen Theorem
  14. Notation
  15. Preliminary Results
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose that the minimization problem (eq:new-min-problem) is considered without additional constraints and $\varrho(\mathcal{X})$ denotes the value function of the problem. If $(\varrho_{a})_{a\in A}$ are risk measures satisfying the Lebesgue property, $(\varrho_{a})_{a\in A}$ satisfies a measurabi for any probability measure $\mathbb{Q}\ll\mathbb{P}$, where $\varrho^{\ast}_{a}$ denotes the conve

Theorems & Definitions (64)

  • Theorem
  • Example 1
  • Example 2: Aumann, aumanncontinuum
  • Example 3: Shapley, shape
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 4
  • ...and 54 more