Model Combination in Risk Sharing under Ambiguity

Paper: arXiv 2504.02987

Authors: Emma Kroell, Sebastian Jaimungal, Silvana M. Pesenti

Abstract

We consider the problem of an agent who faces losses in continuous time over a finite time horizon and may choose to share some of these losses with a counterparty. The agent is uncertain about the true loss distribution and has multiple models for the losses. Their goal is to optimize a mean-variance type criterion with model combination under ambiguity through risk sharing. We construct such a criterion using the chi-squared divergence, adapting the monotone mean-variance preferences of Maccheroni et al. (2009) to the model combination setting and exploit a dual representation to expand the state space, yielding a time consistent problem. Assuming a Cramér-Lundberg loss model, we fully characterize the optimal risk sharing contract and the agent’s wealth process under the optimal strategy. Furthermore, we prove that the strategy we obtain is admissible and that the value function satisfies the appropriate verification conditions. Finally, we apply the optimal strategy to an insurance setting using data from a Spanish automobile insurance portfolio, where we obtain differing models using cross-validation and provide numerical illustrations of the results.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous continuous-time theoretical framework combining chi-squared divergence with monotone mean-variance preferences under ambiguity. While it includes a brief numerical illustration using Spanish automobile insurance data, the contribution is heavily weighted toward advanced mathematical stochastic control and verification proofs rather than extensive empirical backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimize continuous-time risk sharing under model ambiguity"] --> B["Formulate Preferences: Adapt Maccheroni et al. (2009) Mean-Variance using Chi-Squared Divergence"]
    B --> C["Duality & State Expansion: Convert to time-consistent problem"]
    C --> D["Theoretical Analysis: Apply Cramér-Lundberg loss model"]
    E["Data Input: Spanish Automobile Insurance Portfolio (Cross-Validation models)"] --> F["Numerical Application: Implement optimal strategy"]
    D --> F
    F --> G["Outcomes: Fully characterized optimal risk sharing contract & verified admissible wealth process"]

Cited References (55)

1. Limit theorems for stochastic processes (2013) — link
2. Optimal reinsurance designs based on risk measures: a review (2020) — link
3. An old-new concept of convex risk measures: The optimized certainty equivalent (2007) — link
4. Expected utility, penalty functions, and duality in stochastic nonlinear programming (1986) — link
5. Risk minimizing portfolios and HJBI equations for stochastic differential games (2008) — link
6. On the dynamic representation of some time-inconsistent risk measures in a B rownian filtration (2020) — link
7. Stochastic control of optimized certainty equivalents (2022) — link
8. Ambiguity Aversion, Robustness, and the Variational Representation of Preferences (2006)

2 of 55 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.