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

Alexander Erreygers

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'.

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

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'.

Paper Structure

This paper contains 14 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Counting Paths, Variables & Processes
  3. Game-theoretic Upper Expectations
  4. Betting Strategies & Capital Processes for the Poisson Process
  5. Properties of the Conditional Upper Expectation for the Poisson Process
  6. Properties Related to the Rate Bounds
  7. The Strong Markov Property
  8. The Law of Iterated Upper Expectations
  9. Connection to the Sublinear Poisson Semigroup
  10. Conclusion
  13. Relegated proofs
  14. Additional results and relegated proofs for results in sec:game-theoretic upper expectations

Key Result

proposition 1

Suppose $\mathfrak{K}$ contains all constant processes and is a cone---that is, closed under pointwise addition and multiplication with positive scalars. Then for all $\tau\in\mathfrak{T}$, $f, g\in\overline{\mathbb{V}}_{\tau}$ and $\mu\in\mathbb{R}$, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (14)

  • proposition 1
  • proposition 2
  • proposition 3
  • proposition 4
  • proposition 5
  • proof
  • proposition 6
  • proposition 7
  • proof
  • theorem 1
  • ...and 4 more