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Zbl 1133.60016
Random rewards, fractional Brownian local times and stable self-similar processes.
(English)
[J] Ann. Appl. Probab. 16, No. 3, 1432-1461 (2006). ISSN 1050-5164

The authors introduce a new class of self-similar stable processes, namely, the FBM-H-local time fractional symmetric $\alpha$-stable motion, defined by $$Y(t) = \int_{\Omega'} \int_{\Bbb R} l(x, t) (\omega') M(d\omega', dx),$$ where $l(x, t) (\omega')$ is the jointly continuous local time of a fractional Brownian motion with self-similarity index $H \in (0, 1)$ defined on a probability space $(\Omega', {\cal F}', {\Bbb P}')$, and where $M$ is an S$\alpha$S random measure on the space $\Omega'\times {\Bbb R}$ with control measure ${\Bbb P}'\times \text{ Leb}$. \par The authors prove several fundamental properties of $Y$, among them, they show that (i) $Y$ is self-similar with exponent $H' = 1 - H + H/\alpha$ 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 $\vert t-s\vert ^{1 - H} (\log 1/\vert t-s\vert )^{H+1/2}$; (iv) $Y$ can be represented as a series of absolutely continuous self-similar stable processes with the same index $H'$; (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.
[Yimin Xiao (East Lansing)]
MSC 2000:
*60G18 Self-similar processes
60G52 Stable processes
60G17 Sample path properties

Keywords: stable process; self-similar process; stationary process; integral representation; conservative flow; null flow; fractional Brownian motion; local time; random reward; chaos expansion; superposition of scaled inputs; long memory

Cited in: Zbl 1231.60041 Zbl 1214.60020

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