Ruin Probability Analysis in Insurance Risk Models

Ruin probability analysis quantifies the risk that an insurer’s surplus falls below zero, signalling insolvency. Classical frameworks, such as the Cramér–Lundberg model, treat claim arrivals as a Poisson process and focus on closed-form expressions for the ultimate ruin probability. Contemporary models have extended this by incorporating spectrally negative Lévy processes, surplus-dependent premium rates and mechanisms for capital injections or dividend payments. Central to these analyses is the Gerber–Shiu function, which captures joint information on the time of ruin, the deficit at ruin and associated penalty measures. Mathematical techniques range from integro-differential equations and Hamilton–Jacobi–Bellman variational inequalities to scale-function methods and asymptotic approximations. Recent work addresses dependencies in claims via self-exciting processes, periodic observation and multivariate portfolios influenced by common environmental factors. Such developments enhance the realism of risk assessment, inform optimal dividend and capital-injection strategies, and support regulatory solvency requirements. The global significance of these models lies in their capacity to guide pricing, reinsurance arrangements and risk management practices under diverse economic conditions.

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Ruin Probability Analysis in Insurance Risk Models publication trend

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Technical terms

Ruin probability: The probability that an insurer’s surplus falls below zero over a specified horizon.

Surplus process: A stochastic model representing an insurer’s reserve evolution, accounting for premiums collected and claims paid.

Spectrally negative Lévy process: A stochastic process with jumps only in the negative direction, used to model claim shocks.

Gerber–Shiu function: A transform capturing the joint distribution of ruin time, deficit at ruin and penalty costs.

Scale functions (W, Z): Special q-harmonic functions that characterise first-passage probabilities for spectrally negative Lévy processes.

References

1. Optimal dividend policy with self-exciting claims in the Gamma–Omega model. Finance Research Letters (2024).
2. On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums. Journal of Optimization Theory and Applications (2015).
3. The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems. ESAIM Probability and Statistics (2020).
4. Asymptotics and Approximations of Ruin Probabilities for Multivariate Risk Processes in a Markovian Environment. Methodology and Computing in Applied Probability (2019).