Ruin theory is a fundamental concept in actuarial science, particularly relevant for understanding the long-term financial stability of insurance companies. It deals with the probability that an insurer's surplus will fall below zero, leading to insolvency or 'ruin'. This module will explore the core principles of ruin theory, its key models, and the factors that influence the probability of ruin.

At its heart, ruin theory models the evolution of an insurer's surplus over time. The surplus changes based on two primary forces: premiums received (inflows) and claims paid out (outflows). If claims consistently exceed premiums and investment income, the surplus can deplete, eventually leading to ruin.

Several models are used to analyze ruin probability, each with different assumptions about claim arrivals and amounts. The most fundamental is the classical risk model.

This model assumes that premiums are collected at a constant rate $c$ per unit of time. Claims arrive according to a Poisson process with rate $\lambda$, and each claim amount is an independent and identically distributed random variable $X$ with mean $E[X] = \mu$. The surplus $U_t$ at time $t$ can be described by a stochastic differential equation.

The probability of ruin, denoted by $\psi(u)$, is the probability that the surplus $U_t$ will ever fall below zero, given an initial surplus $u$. This probability is a crucial metric for assessing the riskiness of an insurance operation. Factors influencing $\psi(u)$ include the initial surplus, the premium rate, the frequency and severity of claims, and investment returns.

Several factors can significantly impact the likelihood of an insurer becoming ruined:

The safety loading is the excess of the premium rate over the expected claims per unit of time. It represents the buffer an insurer has against adverse claim experience. A positive safety loading is essential for long-term solvency.

Ruin theory extends beyond the classical model to incorporate more realistic scenarios, such as discrete claim arrivals, different claim distributions, and the impact of investment income and expenses. These extensions provide a more nuanced understanding of risk management for insurers.