The Poisson random walk assumes the following equation:

S_{t}=Poisson(m*t+c)

The model Poisson Random Walk can be used, for example, to describe vehicle accident claims made to an insurance company, or cases of a disease for a health authority: as the number of cars increases, the number of car crashes increases correspondingly according to some function; as the pollution level in a city increases, the number of people with respiratory disease increases.

The fractional variation of the series is much bigger on the left panel than that on the right panel. This is because the standard deviation of Poisson counts equals √λ. Thus, the coefficient of variance (std.dev./mean) is 1/√λ*.* which gets smaller as λ gets bigger, meaning that the larger the expected number of events, the smaller the fractional variation one would observe. This property of a Poisson process is very useful to insurance companies: the more people they cover, the more stable their liabilities become, and the less margin they need to cover themselves at a certain risk level. An example of when big is actually better.

The links to the Poisson Random Walk software specific models are provided here: