Levy Processes with finite variance conditioned to avoid an interval

TL;DR

This paper investigates whether Levy processes with finite variance can be conditioned to avoid an interval, characterizing the conditioned process and analyzing its divergence behavior.

Contribution

It demonstrates that Levy processes with finite second moments can be conditioned to avoid an interval and explicitly constructs the conditioned process as an h-transform.

Findings

Conditioned Levy process diverges to ± infinity with positive probability

Explicit h-transform representation in terms of overshoot distributions

Conditioning is possible for Levy processes with finite second moments

Abstract

Conditioning Markov processes to avoid a set is a classical problem that has been studied in many settings. In the present article we study the question if a Levy process can be conditioned to avoid an interval and, if so, the path behavior of the conditioned process. For Levy processes with finite second moments we show that conditioning is possible and identify the conditioned process as an h-transform of the original killed process. The h-transform is explicit in terms of successive overshoot distributions and is used to prove that the conditioned process diverges to plus infinity and minus infinity with positive probabilities.