Ruin theory, sometimes referred to as collective risk theory, is a branch of
actuarial sciencethat studies an insurer's vulnerability to insolvency based on mathematical modeling of the insurer's surplus.
The theory permits the derivation and calculation of many ruin-related measures and quantities, including the probability of ultimate ruin, the distribution of an insurer's surplus immediately prior to ruin, the deficit at the time of ruin, the distribution of the first drop in surplus given that the drop occurs, etc.
It is also considered as an area of
applied probabilitybecause most of the techniques and methodologies adopted in ruin theory are based on the application of stochastic processes. Though most problems in ruin theory stem from real-life actuarial studies, it is the mathematical aspects of ruin theory that have drawn much of the attention from actuarial scientists and probabilists in the past few decades.
The theoretical foundation of ruin theory, known as the classical compound-Poisson risk model in the literature, was introduced in 1903 by the Swedish actuary Filip Lundberg. [Lundberg, F. (1903) Approximerad Framställning av Sannolikehetsfunktionen, Återförsäkering av Kollektivrisker, Almqvist & Wiksell, Uppsala.] The classical model was later extended to relax assumptions about the inter-claim time distribution, the distribution of claim sizes, etc. In most cases, the principal objective of the classical model and its extensions was to calculate the probability of ultimate ruin.
Ruin theory received a substantial boost with the articles of Powers [Powers, M.R. (1995) A theory of risk, return, and solvency, Insurance: Mathematics and Economics 17(2): 101-118.] in 1995 and Gerber and Shiu [Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin, North American Actuarial Journal 2(1): 48-78.] in 1998, which introduced the expected discounted penalty function, a generalization of the probability of ultimate ruin. This fundamental work was followed by a large number of papers in the ruin literature deriving related quantities in a variety of risk models.
Traditionally, an insurer's surplus has been modeled as the result of two opposing cash flows: an incoming cash flow of premium income collected continuously at the rate of , and an outgoing cash flow due to a sequence of insurance claims that are mutually independent and identically distributed with common distribution function . The arrival of claims is assumed to follow a Poisson process with intensity rate , which means that the number of incurred claims at time is governed by a Poisson distribution with mean . Hence, the insurer's surplus at any time is given by
where the insurer's business commences with an initial surplus level under the probability measure .
The central object of Lundberg's model was to investigate the probability that the insurer's surplus level eventually would fall below zero (making the firm bankrupt). This quantity, called the probability of ultimate ruin, is defined as
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