Optimal reinsurance in a dynamic contagion model: comparing self-exciting and externally-exciting risks

Paper: arXiv 2404.11482

Authors: Claudia Ceci, Alessandra Cretarola

Abstract

We investigate the optimal reinsurance problem in a risk model with jump clustering features. This modeling framework is inspired by the concept initially proposed in Dassios and Zhao (2011), combining Hawkes and Cox processes with shot noise intensity models. Specifically, these processes describe self-exciting and externally excited jumps in the claim arrival intensity, respectively. The insurer aims to maximize the expected exponential utility of terminal wealth for general reinsurance contracts and reinsurance premiums. We discuss two different methodologies: the classical stochastic control approach based on the Hamilton-Jacobi-Bellman (HJB) equation and a backward stochastic differential equation (BSDE) approach. In a Markovian setting, differently from the classical HJB-approach, the BSDE method enables us to solve the problem without imposing any requirements for regularity on the associated value function. We provide a Verification Theorem in terms of a suitable BSDE driven by a two-dimensional marked point process and we prove an existence result relaying on the theory developed in Papapantoleon et al. (2018) for stochastic Lipschitz generators. After discussing the optimal strategy for general reinsurance contracts and reinsurance premiums, we provide more explicit results in some relevant cases. Finally, we provide comparison results that highlight the heightened risk stemming from the self-exciting component in contrast to the externally-excited counterpart and discuss the the monotonicity property of the value function.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using Hawkes and Cox processes alongside BSDEs to solve an optimal reinsurance problem under dynamic contagion. While mathematically sophisticated with solid novel insights regarding self-exciting vs. externally-excited risk dynamics, it lacks empirical validation, backtesting, or real-world data calibration.

Research Flowchart

  flowchart TD
    A[Goal: Optimal Reinsurance under Self-Exciting & Externally-Excited Jump Clustering] --> B[Inputs: General Reinsurance Contracts, Premiums & Marked Point Processes]
    B --> C[Method 1: Classical HJB Equation Approach]
    B --> D[Method 2: BSDE Approach with Stochastic Lipschitz Generator]
    C & D --> E[Verification Theorem & Existence Proof for Value Function]
    E --> F[Derive Explicit Optimal Reinsurance Strategies]
    F --> G[Outcomes: Self-Exciting Risks Present Higher Risk Than External Risks & Value Function Monotonicity]

Cited References (23)

1. Quadratic BSDEs with jumps: a fixed-point approach (2015) — link
2. UTILITY MAXIMIZATION WITH RANDOM HORIZON: A BSDE APPROACH (2015) — link
3. Ruin probabilities and aggregrate claims distributions for shot noise Cox processes (2006) — link
4. Exponential utility maximization under partial information (2009) — link
5. Exponential Utility Maximization in an Incomplete Market with Defaults (2011) — link
6. Spectra of Some Self-Exciting and Mutually Exciting Point Processes (1971) — link
7. Exponential inequalities for ruin probabilities in the Cox case (1988) — link
8. Optimal control of risk exposure, reinsurance and investments for insurance portfolios (2004) — link

0 of 23 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.