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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 $$ Y(t) = \int_{\Omega'} \int_{\Bbb R} l(x, t) (\omega') M(d\omega', dx), $$ 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 ${\Bbb P}'\times \text{ Leb}$. 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.

MSC:
60G18Self-similar processes
60G52Stable processes
60G17Sample path properties
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References:
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