The Poisson Process: From Theory to Insurance Claims

The Homogeneous Poisson Process

The Poisson process is the most fundamental model for counting events that occur randomly over time. A homogeneous Poisson process with rate lambda has three key properties: the number of events in any interval of length t follows a Poisson distribution with mean lambda times t, counts in non-overlapping intervals are independent, and events do not occur simultaneously. In insurance, this process models claim arrivals, where lambda represents the expected number of claims per unit time. The exponential distribution of inter-arrival times follows directly from these assumptions.

Extensions for Insurance

Real insurance claim patterns often violate the constant-rate assumption. The non-homogeneous Poisson process allows the rate to vary over time, capturing seasonal patterns in claims (such as higher auto accident rates in winter). The compound Poisson process pairs random claim arrival times with random claim amounts, forming the basis of aggregate loss models. The mixed Poisson process introduces a random rate parameter, naturally producing overdispersion (variance exceeding the mean) commonly observed in insurance data. These extensions appear extensively in loss modeling and ratemaking.