Ruin Theory: Classical and Modern Approaches

The Classical Model

Ruin theory studies the probability that an insurer's surplus falls below zero. The Cramer-Lundberg model represents the surplus process as initial capital plus premium income (linear in time) minus a compound Poisson process of claims. The probability of ultimate ruin in this model depends on the initial surplus, the safety loading, and the claim size distribution. For exponentially distributed claims, an exact closed-form solution exists. For general distributions, the Pollaczek-Khinchine formula expresses the ruin probability using the claim size distribution's properties, and the Cramer-Lundberg bound provides an exponential upper bound through the adjustment coefficient.

Modern Extensions

Contemporary ruin theory extends the classical model in several directions. The Sparre Andersen model relaxes the Poisson assumption for claim arrivals. Perturbed risk models add a diffusion component to capture small, frequent fluctuations. Levy process models allow for more general jump structures. The Gerber-Shiu discounted penalty function unifies the analysis of ruin time, deficit at ruin, and surplus before ruin into a single framework. These modern tools provide more realistic assessments of insurer solvency risk and inform regulatory capital requirements.