Time Change Equations for L\'evy Type Processes

TL;DR

This paper investigates time change equations for Le9vy-type processes, establishing a link with initial value problems to analyze their properties, existence, and uniqueness, with applications to processes defined by specific symbols.

Contribution

It introduces a novel connection between TCEs and IVPs for Le9vy-type processes, providing general existence and uniqueness results.

Findings

Established a link between TCEs and classical IVPs.

Proved existence and uniqueness of solutions for TCEs.

Applied results to processes with specific symbols.

Abstract

In this paper we analyse time change equations (TCEs) for L\'evy-type processes in detail. To this end we establish a connection between TCEs and classical one-dimensional initial value problems (IVPs) which are easier to handle. Properties of the IVPs are linked with properties of the TCEs. We show in a general setting existence and uniqueness of solutions of the TCEs. Our main result is based on the general path properties for L\'evy-type processes found in Schnurr (2013). Applications include an existence result for processes which correspond to a certain class of given symbols.