Allowing for imprecision in the game-theoretic characterisation of the Poisson process

TL;DR

This paper introduces an imprecise Poisson process by relaxing the assumption of a known rate, using bounds instead, which enhances the realism of game-theoretic process modeling.

Contribution

It generalizes the game-theoretic Poisson process framework to incorporate rate bounds, making the model more flexible and realistic.

Findings

The imprecise Poisson process retains key properties of the standard process.

Using bounds on the rate allows for modeling uncertainty more effectively.

The approach bridges the gap between precise and imprecise stochastic modeling.

Abstract

In their 1993 paper 'Forecasting point and continuous processes: Prequential analysis' in Test, Vovk put forward a game-theoretic definition of the Poisson process. A key assumption therein is that the rate of the Poisson process is known or specified exactly. In contrast, I replace this assumption with the less stringent -- and arguably more realistic -- one that the available information about the process takes the form of bounds on the rate rather than a single, exact value. The resulting process has properties similar to the standard, 'precise' Poisson process, albeit with an imprecise flavour to them, thus justifying the moniker 'imprecise Poisson process'.