Increasing processes and the change of variables formula for non-decreasing functions

TL;DR

This paper characterizes non-decreasing functions that transform increasing processes into pure-jump processes, with applications to subordinators and their generators, advancing understanding of stochastic process transformations.

Contribution

It provides a characterization of functions that map increasing processes to pure-jump processes, including those with bounded variation in the context of subordinators.

Findings

Functions with bounded variation belong to the domain of the extended generator of certain subordinators.

The paper offers a new change of variables formula for non-decreasing functions in stochastic processes.

It characterizes right-continuous non-decreasing functions that produce pure-jump processes from increasing processes.

Abstract

Given an increasing process $(A_t)_{t\geq 0}$, we characterize the right-continuous non-decreasing functions $f: \R_+\to \R_+$ that map $A$ to a pure-jump process. As an example of application, we show for instance that functions with bounded variations belong to the domain of the extended generator of any subordinators with no drift and infinite L\'evy measure.