Conditioning subordinators embedded in Markov processes

TL;DR

This paper explores how subordinators within Markov processes can be conditioned to stay within a specified range, extending known results about Levy processes and their ladder height subordinators.

Contribution

It introduces a general framework for conditioning subordinators embedded in Markov processes to remain in a strip, broadening the understanding of path decompositions.

Findings

Established a method for conditioning subordinators in Markov processes

Extended the concept of staying positive to more general subordinators

Provided insights into the structure of path decompositions

Abstract

The running infimum of a Levy process relative to its point of issue is know to have the same range that of the negative of a certain subordinator. Conditioning a Levy process issued from a strictly positive value to stay positive may therefore be seen as implicitly conditioning its descending ladder heigh subordinator to remain in a strip. Motivated by this observation, we consider the general problem of conditioning a subordinator to remain in a strip. Thereafter we consider more general contexts in which subordinators embedded in the path decompositions of Markov processes are conditioned to remain in a strip.