*(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

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

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 ${|t-s|}^{1-H}{(log1/|t-s\left|\right)}^{H+1/2}$; (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.