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Optimal reinsurance in a competitive market

Lea Enzi, Stefan Thonhauser

TL;DR

The paper develops a rigorous framework for an optimal reinsurance problem in a competitive market by modeling two insurers as a zero-sum stochastic differential game with surplus processes driven by claims in a piecewise deterministic Markov setup. It establishes a dynamic programming principle for the upper and lower value functions and proves that these values are unique viscosity solutions to the corresponding Bellman-Isaacs equations, supported by a thorough comparison theorem. Numerically, it implements policy-iteration schemes on discretized grids to compute Nash-like equilibria under various reinsurance regimes (proportional and excess-of-loss) and claim distributions, illustrating strategic behavior across boundary regions. The work advances practical insights into competitive reinsurance decisions and provides a computational approach to equilibrium analysis in ruin-theoretic PDMP models.

Abstract

We study a stochastic differential game in a ruin theoretic environment. In our setting two insurers compete for market share, which is represented by a joint performance functional. Consequently, one of the insurers strives to maximize it, while the other seeks to minimize it. As a modelling basis we use classical surplus processes extended by dynamic reinsurance opportunities, which allows us to use techniques from the theory of piecewise deterministic Markov processes to analyze the resulting game. In this context we show that a dynamic programming principle for the upper and lower value of the game holds true and that these values are unique viscosity solutions to the associated Bellman-Isaacs equations. Finally, we provide some numerical illustrations.

Optimal reinsurance in a competitive market

TL;DR

The paper develops a rigorous framework for an optimal reinsurance problem in a competitive market by modeling two insurers as a zero-sum stochastic differential game with surplus processes driven by claims in a piecewise deterministic Markov setup. It establishes a dynamic programming principle for the upper and lower value functions and proves that these values are unique viscosity solutions to the corresponding Bellman-Isaacs equations, supported by a thorough comparison theorem. Numerically, it implements policy-iteration schemes on discretized grids to compute Nash-like equilibria under various reinsurance regimes (proportional and excess-of-loss) and claim distributions, illustrating strategic behavior across boundary regions. The work advances practical insights into competitive reinsurance decisions and provides a computational approach to equilibrium analysis in ruin-theoretic PDMP models.

Abstract

We study a stochastic differential game in a ruin theoretic environment. In our setting two insurers compete for market share, which is represented by a joint performance functional. Consequently, one of the insurers strives to maximize it, while the other seeks to minimize it. As a modelling basis we use classical surplus processes extended by dynamic reinsurance opportunities, which allows us to use techniques from the theory of piecewise deterministic Markov processes to analyze the resulting game. In this context we show that a dynamic programming principle for the upper and lower value of the game holds true and that these values are unique viscosity solutions to the associated Bellman-Isaacs equations. Finally, we provide some numerical illustrations.

Paper Structure

This paper contains 13 sections, 6 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model setup
  3. The underlying insurance model
  4. A two players game
  5. A-priori properties of the value functions
  6. Characterization of the value functions
  7. Viscosity solutions to Bellmann-Isaacs equations
  8. Uniqueness of solution to Bellman-Isaacs equation
  9. Numerical examples
  10. Proportional reinsurance, variance principle and exponential claims
  11. Excess of loss reinsurance and expectation principle
  12. Exponential claims
  13. Pareto type II distributed claims

Key Result

Lemma 3.1

The function $t\mapsto J^{u_1 u_2}(\phi^{u_1u_2}(t,x))$ is bounded and continuous for any controls $u_1 \in \mathcal{U}_1$ and $u_2 \in \mathcal{U}_2$.

Figures (2)

  • Figure 1: Strategies of player 1 and 2 after 10 iterations. The controls of Player 1 and 2 correspond to the solid and dashed line respectively.
  • Figure 2: Strategies of player 1 and 2 after 10 iterations with exponentially distributed claims. The solid line represents player one and the dashed player two. Technically, "no reinsurance" corresponds to an infinite value of the control, but for illustrative purposes it is shown in gray.

Theorems & Definitions (16)

  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Theorem 3.1: Dynamic programming principle
  • ...and 6 more