Paper: arXiv 2512.13562

Authors: Christian Furrer, Philipp C. Hornung

Abstract

The classic semi-Markov disability model is expanded with individual and collective health claims to improve its explanatory and predictive power – in particular in the context of group experience rating. The inclusion of collective health claims leads to a computationally challenging many-body problem. By adopting a mean-field approach, this many-body problem can be approximated by a non-linear one-body problem, which in turn leads to a transparent pricing method for disability coverages based on a lower-dimensional system of non-linear forward integro-differential equations. In a practice-oriented simulation study, the mean-field approximation clearly stands its ground in comparison to naïve Monte Carlo methods.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous theoretical formulation using mean-field limits for semi-Markov processes with interaction via health claims. While mathematically complex with formal integro-differential proofs, its empirical section is primarily based on simulated data rather than extensive empirical market datasets.

Research Flowchart

  flowchart TD
    A[Goal: Price disability insurance with individual & collective health claims] --> B[Model: Expand semi-Markov model to include collective claims]
    B --> C[Method: Apply Mean-Field Approach to approximate many-body problem]
    C --> D[Inputs: Group policyholder data & health claim distributions]
    D --> E[Computation: Solve system of non-linear forward integro-differential equations]
    E --> F[Validation: Perform practice-oriented Monte Carlo simulation study]
    F --> G[Outcome: Transparent, low-dimensional pricing method that outperforms naïve Monte Carlo]

Cited References (13)

  1. Dynamic modelling and causality (1987) — link
  2. Cash flows and policyholder behaviour in the semi-Markov life insurance setup (2015) — link
  3. Multistate models in health insurance (2012) — link
  4. Nonlinear reserving in life insurance: Aggregation and mean-field approximation (2016) — link
  5. On solutions of Kolmogorov’s equations for nonhomogeneous jump Markov processes (2014) — link
  6. Kolmogorov’s equations for jump Markov processes with unbounded jump rates (2022) — link
  7. Experience rating in the classic Markov chain life insurance setting: An empirical Bayes and multivariate frailty approach (2019) — link
  8. Bivariate phase-type distributions for experience rating in disability insurance (2025) — link
and 5 more
  1. Durational effects and non-smooth semi- M arkov models in life insurance (2008)
  2. Inhomogeneous S emi- M arkov P rocesses, S elect A ctuarial T ables, and D uration- D ependence in D emography (1972) — link
  3. Mean-field approximations in insurance — → 2511.04198
  4. Point process theory and applications: Marked point and piecewise deterministic processes (2006) — link
  5. Simulation of nonhomogeneous Poisson processes by thinning (1979)

1 of 13 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.