On Rd-valued multi-self-similar Markov processes

Loïc Chaumont and Salem Lamine

Stochastic Processes and their Applications, 2020, vol. 130, issue 5, 3174-3192

Abstract: An Rd-valued Markov process Xt(x)=(Xt1,x1,…,Xtd,xd), t≥0,x∈Rd is said to be multi-self-similar with index (α1,…,αd)∈[0,∞)d if the identity in law (ciXti,xi∕ci,t≥0)1≤i≤d=(d)(Xcαt(x),t≥0),where cα=∏i=1dciαi, is satisfied for all c1,…,cd>0 and all starting point x. Multi-self-similar Markov processes were introduced by Jacobsen and Yor (2003) in the aim of extending the Lamperti transformation of positive self-similar Markov processes to R+d-valued processes. This paper aims at giving a complete description of all Rd-valued multi-self-similar Markov processes. We show that their state space is always a union of open orthants with 0 as the only absorbing state and that there is no finite entrance law at 0 for these processes. We give conditions for these processes to satisfy the Feller property. Then we show that a Lamperti-type representation is also valid for Rd-valued multi-self-similar Markov processes. In particular, we obtain a one-to-one relationship between this set of processes and the set of Markov additive processes with values in {−1,1}d×Rd. We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes.

Keywords: Multi-self-similarity; Markov additive process; Lévy process; Time change (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1016/j.spa.2019.09.009

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