In actuarial science and applied probability, ruin theory (sometimes risk theory or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.

The theoretical foundation of ruin theory, known as the Cramér–Lundberg model (or classical compound-Poisson risk model, classical risk process or Poisson risk process) was introduced in 1903 by the Swedish actuary Filip Lundberg. Lundberg's work was republished in the 1930s by Harald Cramér.

The model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims. Premiums arrive a constant rate ${\textstyle c>0}$ from customers and claims arrive according to a Poisson process ${\displaystyle N_{t}}$ with intensity ${\textstyle \lambda }$ and are independent and identically distributed non-negative random variables ${\displaystyle \xi _{i}}$ with distribution ${\textstyle F}$ and mean ${\textstyle \mu }$ (they form a compound Poisson process). So for an insurer who starts with initial surplus ${\textstyle x}$, the aggregate assets ${\displaystyle X_{t}}$ are given by:

The central object of the model is to investigate the probability that the insurer's surplus level eventually falls below zero (making the firm bankrupt). This quantity, called the probability of ultimate ruin, is defined as

where the time of ruin is ${\displaystyle \tau =\inf\{t>0\,:\,X(t)<0\}}$ with the convention that ${\displaystyle \inf \varnothing =\infty }$. This can be computed exactly using the Pollaczek–Khinchine formula as (the ruin function here is equivalent to the tail function of the stationary distribution of waiting time in an M/G/1 queue)

where ${\displaystyle F_{l}}$ is the transform of the tail distribution of ${\displaystyle F}$,

and ${\displaystyle \cdot ^{\ast n}}$ denotes the ${\displaystyle n}$-fold convolution. In the case where the claim sizes are exponentially distributed, this simplifies to

E. Sparre Andersen extended the classical model in 1957 by allowing claim inter-arrival times to have arbitrary distribution functions.