Abstract
Since the seminal work of Lamperti, there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-similar Markov processes (rssMps), there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti- Kiu representation to the origin.
| Original language | English |
|---|---|
| Pages (from-to) | 1952-2003 |
| Number of pages | 52 |
| Journal | Annals of Probability |
| Volume | 45 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 1 May 2017 |
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Keywords
- Fluctuation theory
- Markov additive process
- Self-similar process
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty
Cite this
Real self-similar processes started from the origin. / Dereich, Steffen; Döring, Leif; Kyprianou, Andreas E.
In: Annals of Probability, Vol. 45, No. 3, 01.05.2017, p. 1952-2003.Research output: Contribution to journal › Article
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TY - JOUR
T1 - Real self-similar processes started from the origin
AU - Dereich, Steffen
AU - Döring, Leif
AU - Kyprianou, Andreas E.
PY - 2017/5/1
Y1 - 2017/5/1
N2 - Since the seminal work of Lamperti, there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-similar Markov processes (rssMps), there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti- Kiu representation to the origin.
AB - Since the seminal work of Lamperti, there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-similar Markov processes (rssMps), there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti- Kiu representation to the origin.
KW - Fluctuation theory
KW - Markov additive process
KW - Self-similar process
UR - http://www.scopus.com/inward/record.url?scp=85019177216&partnerID=8YFLogxK
UR - http://dx.doi.org/10.1214/16-AOP1105
U2 - 10.1214/16-AOP1105
DO - 10.1214/16-AOP1105
M3 - Article
VL - 45
SP - 1952
EP - 2003
JO - Annals of Probability
JF - Annals of Probability
SN - 0091-1798
IS - 3
ER -