### 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

*Annals of Probability*,

*45*(3), 1952-2003. https://doi.org/10.1214/16-AOP1105

**Real self-similar processes started from the origin.** / Dereich, Steffen; Döring, Leif; Kyprianou, Andreas E.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 45, no. 3, pp. 1952-2003. https://doi.org/10.1214/16-AOP1105

}

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 -