Conditioned real self-similar Markov processes

TL;DR

This paper extends the concept of conditioning stable processes to avoid or absorb at the origin within the broader class of real self-similar Markov processes, using transformations related to Markov additive processes.

Contribution

It generalizes previous conditioning results for stable processes to all real self-similar Markov processes, linking these conditionings to Cramér-Esscher transforms of the underlying Markov additive processes.

Findings

Conditioning to avoid the origin corresponds to a Cramér-Esscher transform of the MAP.

Conditioning to absorb at the origin is characterized within the same framework.

Results apply to stable processes with different stability indices, including \\alpha in (0,1) and (1,2).

Abstract

In recent work, Chaumont et al. [9] showed that is possible to condition a stable process with index ${\alpha} \in (1,2)$ to avoid the origin. Specifically, they describe a new Markov process which is the Doob h-transform of a stable process and which arises from a limiting procedure in which the stable process is conditioned to have avoided the origin at later and later times. A stable process is a particular example of a real self-similar Markov process (rssMp) and we develop the idea of such conditionings further to the class of rssMp. Under appropriate conditions, we show that the specific case of conditioning to avoid the origin corresponds to a classical Cram\'er-Esscher-type transform to the Markov Additive Process (MAP) that underlies the Lamperti-Kiu representation of a rssMp. In the same spirit, we show that the notion of conditioning a rssMp to continuously absorb at the origin also fits the same mathematical framework. In particular, we characterise the stable process conditioned to continuously absorb at the origin when ${\alpha} \in(0,1)$. Our results also complement related work for positive self-similar Markov processes in [10].