Mean-field approximations in insurance

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.

• The paper establishes convergence of insurance liabilities to mean-field models using propagation of chaos theory.
• It derives non-linear forward equations and quantifies convergence rates through Wasserstein metrics and Grönwall-type bounds.
• It provides a scalable framework for computing reserves and expected claims in both life and non-life insurance.

Expert Summary of "Mean-field approximations in insurance" (2511.04198)

Overview and Motivation

Insurance liability modeling often faces computational bottlenecks due to dependencies among individuals within a cohort. Exact calculations require solving high-dimensional, coupled systems of linear forward integro-differential equations. The paper rigorously explores mean-field approximations as a scalable alternative. By replacing collective state dependencies with expectations, these approximations reduce the dimensionality, resulting in low-dimensional non-linear forward equations. The manuscript establishes convergence results, validates mean-field strategies for actuarial applications, and leverages propagation of chaos theory (as initially developed by Kac and McKean) adapted for jump processes.

Mathematical Foundations

The primary mathematical architecture consists of stochastic jump processes with state- and measure-dependent intensities. Individuals' liabilities are expressed as expectations (possibly conditional) of functionals of these paths. The paper introduces:

• Non-linear jump processes (McKean–Vlasov type): The intensity kernel depends on the distribution of the process itself, leading to non-linear semigroup equations for marginals.
• Occupation and transition probabilities: Forward equations governing these are derived for both standard and distribution-dependent SDEs.
• Mean-field approximation: Empirical distributions are replaced by process marginals, leading to mean-field models.
• Chaos/Propagation of Chaos: Weak convergence of joint distributions of subcohorts to independent distributions (law of large numbers effect remains despite dependency).

The technical results cover existence, uniqueness, and well-posedness for SDEs with measure-dependent jump kernels, and establish the equivalence between jump-size and jump-destination specifications.

Main Theoretical Results

• Convergence of Mean-field Liabilities: Under regularity (continuity, uniform integrability), insurance liabilities defined as expectations of process functionals converge to their mean-field analog as cohort size increases.
• Propagation of Chaos: The joint law for any finite subcohort converges weakly to a product measure corresponding to independent mean-field jump processes. Wasserstein distance is used to quantify convergence.
• Conditional Results: For countable state spaces and under chaotic initial state conditions, conditional distributions also exhibit chaos and convergence to mean-field limits.
• Practical Calculation: Occupation probabilities and reserves in the mean-field model are computed by solving non-linear forward equations, bypassing the exponential growth in dimensionality encountered in direct modeling of the cohort.

Applications in Insurance

Non-life Insurance

• Claim Amounts: Individual claim processes with occurrence intensity and claim sizes allowed to depend on cohort-wide statistics (e.g., average claim size).
• Convergence: Expected claim amounts of individuals (both unconditional and conditional on covariates) converge to those from mean-field models.
• Law of Large Numbers and CLT: The portfolio average of claims converges in $L^2$ to the mean-field expected claim; central limit theorem holds under stronger moment and decorrelation conditions.
• Implementational Guidance: Sufficient conditions on claim intensity and size distributions for regularity are stated, including applications for measure-dependent Gamma-distributed claim sizes.

Life Insurance

• Biometric Risks: Cohort modeled as jump processes; payment streams include sojourn and transition payments.
• Reserves: Calculation of portfolio and state-wise reserves are simplified via mean-field approximations.
• Epidemic Risk: Mean-field intensity for infection in SIRD model is given by the marginal probability in the mean-field process, instead of the empirical fraction infected.

General Framework

• Exchangeability: Symmetry in dependencies is crucial for chaos propagation.
• Conditional Convergence: Extends to covariate-dependent models (e.g., disability insurance conditional on health insurance claims).
• Numerical Implementation: Forward (not backward) methods are recommended due to non-linearity and measure-dependence.

Numerical and Empirical Results

While explicit numerical experiments are not reported, the paper provides rigorous convergence rates using Wasserstein metrics and bounds derived via Grönwall-type inequalities. The theoretical machinery ensures that practical actuarial scenarios (large, dependent cohorts) can reliably use mean-field models provided the regularity and chaos assumptions are met.

Implications and Future Directions

The results justify the actuarial use of mean-field approximations for large dependent portfolios, supporting practical computation of reserves, expected claims, and risk measures. The propagation of chaos results reinforce the diversification principle in insurance, demonstrating that dependency in intensities does not inhibit law-of-large-numbers effects.

Potential future directions include:

• Generalizing to non-homogeneous or network-structured cohorts: Extensions to spatial contagion models or complex dependency networks.
• Numerical schemes for non-linear forward equations: Robust algorithms to guarantee uniqueness and stability.
• Rigorous validation of CLT decorrelation rates: Especially for strongly dependent or heavy-tailed claims.

The theoretical framework can further inform risk management practices, such as dynamic solvency modeling and stress-testing scenarios, where collective effects are significant.

Conclusion

The paper provides a comprehensive mathematical foundation for mean-field approximations in insurance liability modeling for dependent cohorts. It establishes conditions and rates for propagation of chaos, guaranteeing the validity of scalable mean-field models for a range of actuarial applications in both life and non-life insurance. The results enable accurate reserving, portfolio risk assessment, and model simplification without loss of theoretical rigor, provided the system is sufficiently regular and chaotic. These advances offer practical tools for actuaries confronted with computationally prohibitive dependent risk models.