Riemann surfaces for integer counting processes

TL;DR

This paper explores the connection between integer counting processes and complex algebraic curves, showing how their probabilities can be represented as contour integrals on associated Riemann surfaces, with detailed examples.

Contribution

It introduces a novel geometric framework linking counting processes to Riemann surfaces via their generator's characteristic equation.

Findings

Probability expressed as contour integral on Riemann surface

Establishment of a geometric interpretation for counting process probabilities

Detailed examples illustrating the theoretical framework

Abstract

Integer counting processes increment of an integer value at transitions between states of an underlying Markov process. The generator of a counting process, which depends on a parameter conjugate to the increments, defines a complex algebraic curve through its characteristic equation, and thus a compact Riemann surface. We show that the probability of a counting process can then be written as a contour integral on that Riemann surface. Several examples are discussed in details.