Reciprocal class of jump processes

TL;DR

This paper characterizes the reciprocal class of compound Poisson processes with finite jumps, using geometric and probabilistic tools to identify when two processes share the same bridges and how to distinguish their invariants.

Contribution

It introduces a novel characterization of the reciprocal class for jump processes via reciprocal invariants and geometric analysis, providing explicit conditions for process equivalence.

Findings

Characterization of reciprocal class using reciprocal invariants.

Explicit criteria for process equivalence within the class.

Interpretation of invariants as short-time cycle probabilities.

Abstract

Processes having the same bridges as a given reference Markov process constitute its {\it reciprocal class}. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set $\mathcal{A} \subset \mathbb{R}^d$. We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the \textit{reciprocal invariants}. The geometry of $\mathcal{A}$ plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.