# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Random rewards, fractional Brownian local times and stable self-similar processes. (English) Zbl 1133.60016

The authors introduce a new class of self-similar stable processes, namely, the FBM-H-local time fractional symmetric $\alpha$-stable motion, defined by

$Y\left(t\right)={\int }_{{{\Omega }}^{\text{'}}}{\int }_{ℝ}l\left(x,t\right)\left({\omega }^{\text{'}}\right)M\left(d{\omega }^{\text{'}},dx\right),$

where $l\left(x,t\right)\left({\omega }^{\text{'}}\right)$ is the jointly continuous local time of a fractional Brownian motion with self-similarity index $H\in \left(0,1\right)$ defined on a probability space $\left({{\Omega }}^{\text{'}},{ℱ}^{\text{'}},{ℙ}^{\text{'}}\right)$, and where $M$ is an S$\alpha$S random measure on the space ${{\Omega }}^{\text{'}}×ℝ$ with control measure ${ℙ}^{\text{'}}×\phantom{\rule{4.pt}{0ex}}\text{Leb}$.

The authors prove several fundamental properties of $Y$, among them, they show that (i) $Y$ is self-similar with exponent ${H}^{\text{'}}=1-H+H/\alpha$ and has stationary increments; (ii) the corresponding stable noise of $Y$ is generated by a conservative null flow; (iii) the uniform modulus of continuity of $Y$ is at most ${|t-s|}^{1-H}{\left(log1/|t-s|\right)}^{H+1/2}$; (iv) $Y$ can be represented as a series of absolutely continuous self-similar stable processes with the same index ${H}^{\text{'}}$; (v) When $H=1/2$, $Y$ is the limiting process of the random reward scheme. In the last section of the paper, the authors discuss possible extensions of their model and some unsolved problems.

##### MSC:
 60G18 Self-similar processes 60G52 Stable processes 60G17 Sample path properties