Markov Process Jump Times and Their Cox Construction

TL;DR

This paper reveals that the Cox construction of totally inaccessible stopping times and the jump times of Feller processes are fundamentally the same, linking these concepts to predictable stopping times and their hitting times.

Contribution

It demonstrates the equivalence between Cox construction-based stopping times and Feller process jump times, unifying these concepts within stochastic process theory.

Findings

Cox construction and Feller process jump times are equivalent.

Predictable stopping times can be represented as hitting times of zero.

Provides a unified view of inaccessible and predictable stopping times.

Abstract

In this short paper, we connect the procedure of constructing a totally inaccessible stopping time for a given process using the well-known Cox construction, dependent on an independent exponential random variable; with naturally occurring jump times of Feller processes. Ultimately, we show that these two phenomena are not only related, but are in fact two examples of the same object. We link this fact to the behaviour of predictable stopping times, by proving that they can always be written as the hitting time of zero of a continuous process.