CramérLundberg

The Cramér-Lundberg model, also known as the collective risk model, is a fundamental mathematical model used in actuarial science to describe the ruin probability of an insurance company. It was developed by Harald Cramér and Filip Lundberg in the early 20th century. The model assumes that the insurance company's surplus changes over time due to two main processes: the arrival of claims and the investment income generated by the surplus.

In the standard Cramér-Lundberg model, claims arrive according to a Poisson process. The amount of each claim is a random variable, typically assumed to be exponentially distributed or from a more general distribution. Premiums are collected at a constant rate, which is assumed to be greater than the expected rate of claims. The surplus at any given time is the initial surplus plus the accumulated premiums and investment income, minus the total amount of claims paid out.

The primary objective of studying the Cramér-Lundberg model is to determine the probability of ruin, which is the event that the surplus ever falls below zero. This probability depends on various factors, including the initial surplus, the premium rate, the claim arrival rate, and the distribution of claim amounts. Cramér's work established that under certain conditions, the ruin probability can be expressed using a formula related to the adjustment coefficient, a key parameter that measures the rate at which the surplus grows relative to the risk. Lundberg's contributions focused on refining the model and its applications. The model has been extended in numerous ways to account for more complex scenarios, such as different claim arrival processes, time-dependent premiums, and reinvestment strategies.