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Mean-field approximations in insurance

Philipp C. Hornung

TL;DR

The paper tackles the problem of computing insurance liabilities for cohorts with dependent individuals, where forward equations scale poorly with cohort size. It develops a mean-field framework based on distribution-dependent jump processes, proving propagation of chaos and conditional chaos: as the cohort size $n$ grows, the joint laws converge to products of nonlinear mean-field laws, and the cohort-level quantities converge to corresponding mean-field counterparts. By formulating forward equations for the mean-field system, the authors provide a scalable method to compute liabilities, reserves, and present values in both non-life and life insurance contexts, including epidemic-risk settings. The results preserve diversification effects in large portfolios and furnish LLNs and CLTs under suitable moment conditions, offering practical tools for pricing and reserving in large, potentially contagious or networked risk environments.

Abstract

The calculation of the insurance liabilities of a cohort of dependent individuals in general requires the solution of a high-dimensional system of coupled linear forward integro-differential equations, which is infeasible for a larger cohort. However, by using a mean-field approximation, the high dimensional system of linear forward equations can be replaced by a low-dimensional system of non-linear forward integro-differential equations. We show that, subject to certain regularity conditions, the insurance liability viewed as a (conditional) expectation of a functional of an underlying jump process converges to its mean-field approximation, as the number of individuals in the cohort goes to infinity. Examples from both life- and non-life insurance illuminate the practical importance of mean-field approximations.

Mean-field approximations in insurance

TL;DR

The paper tackles the problem of computing insurance liabilities for cohorts with dependent individuals, where forward equations scale poorly with cohort size. It develops a mean-field framework based on distribution-dependent jump processes, proving propagation of chaos and conditional chaos: as the cohort size grows, the joint laws converge to products of nonlinear mean-field laws, and the cohort-level quantities converge to corresponding mean-field counterparts. By formulating forward equations for the mean-field system, the authors provide a scalable method to compute liabilities, reserves, and present values in both non-life and life insurance contexts, including epidemic-risk settings. The results preserve diversification effects in large portfolios and furnish LLNs and CLTs under suitable moment conditions, offering practical tools for pricing and reserving in large, potentially contagious or networked risk environments.

Abstract

The calculation of the insurance liabilities of a cohort of dependent individuals in general requires the solution of a high-dimensional system of coupled linear forward integro-differential equations, which is infeasible for a larger cohort. However, by using a mean-field approximation, the high dimensional system of linear forward equations can be replaced by a low-dimensional system of non-linear forward integro-differential equations. We show that, subject to certain regularity conditions, the insurance liability viewed as a (conditional) expectation of a functional of an underlying jump process converges to its mean-field approximation, as the number of individuals in the cohort goes to infinity. Examples from both life- and non-life insurance illuminate the practical importance of mean-field approximations.

Paper Structure

This paper contains 18 sections, 41 theorems, 272 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Jump processes
  3. Jump processes
  4. Distribution dependent jump process
  5. Jump destination specification
  6. Mean-field approximation
  7. Proof of Theorem \ref{['th:SDE_Chaos']}
  8. Mean-field approximation of the conditional distribution
  9. Proof of Theorem \ref{['th:SDE_Chaos-cond']}
  10. Non-life insurance applications
  11. Life insurance applications
  12. Proof of Theorems \ref{['th:SDE-cond']} and \ref{['th:DDSDE-cond']}
  13. Proof of Theorem \ref{['th:SDE-cond']}
  14. Proof of Theorem \ref{['th:DDSDE-cond']}
  15. Proof of Theorem \ref{['th:DDSDE-existence']}
  16. ...and 3 more sections

Key Result

Theorem 2.1

Assume that the function $\lambda:[0,T]\times E\rightarrow [0,\infty)$ is Borel-measurable and that there exists a $C_{\lambda},C_r>0$ such that Then there exists a unique weak solution to the SDE (eq:SDE), which is non-explosive.

Figures (1)

  • Figure 1: State space $E=\{1,2,3,4\}$ for the SIRD model. The arrows represent the possible transitions.

Theorems & Definitions (101)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • ...and 91 more