Paper: arXiv 2504.19151

Authors: Hansjoerg Albrecher, Pablo Azcue, Nora Muler

Abstract

We study optimal dividend strategies for an insurance company facing natural catastrophe claims, anticipating the arrival of a climate tipping point after which the claim intensity and/or the claim size distribution of the underlying risks deteriorates irreversibly. Extending earlier literature based on a shot-noise Cox process assumption for claim arrivals, we show that the non-stationary feature of such a tipping point can, in fact, be an advantage for shareholders seeking to maximize expected discounted dividends over the lifetime of the portfolio. Assuming the tipping point arrives according to an Erlang distribution, we demonstrate that the corresponding system of two-dimensional stochastic control problems admits a viscosity solution, which can be numerically approximated using a discretization of the current surplus and the claim intensity level. We also prove uniform convergence of this discrete solution to that of the original continuous problem. The results are illustrated through several numerical examples, and the sensitivity of the optimal dividend strategies to the presence of a climate tipping point is analyzed. In all these examples, it turns out that when the insurance premium is adjusted fairly at the moment of the tipping point, and all quantities are observable, the non-stationarity introduced by the tipping point can, in fact, represent an upward potential for shareholders.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework using two-dimensional stochastic control, Hamilton-Jacobi-Bellman equations, and viscosity solutions to address optimal dividend strategies during a climate tipping point. However, it relies heavily on stylized numerical illustrations rather than empirical backtesting with real actuarial datasets.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Maximize dividends for NatCat insurer facing a climate tipping point"] --> B["Modeling Inputs:<br>Shot-noise Cox process (claims), Erlang distribution (tipping point timing)"]
    B --> C["Methodology:<br>Formulate non-stationary 2D stochastic control problem"]
    C --> D["Computational Process:<br>Grid discretization of surplus & claim intensity"]
    D --> E["Theoretical Proof:<br>Viscosity solution & uniform convergence of discrete scheme"]
    E --> F["Numerical Analysis:<br>Sensitivity tests & fair premium adjustments at tipping point"]
    F --> G["Key Finding:<br>Tipping point non-stationarity offers upward potential for shareholders"]