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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 \[ Y(t) = \int_{\Omega'} \int_{\mathbb R} l(x, t) (\omega') M(d\omega', dx), \] 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 \((\Omega', {\mathcal F}', {\mathbb P}')\), and where \(M\) is an S\(\alpha\)S random measure on the space \(\Omega'\times {\mathbb R}\) with control measure \({\mathbb P}'\times \text{ Leb}\).
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 \(| t-s| ^{1 - H} (\log 1/| t-s| )^{H+1/2}\); (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:

60G18 Self-similar stochastic processes
60G52 Stable stochastic processes
60G17 Sample path properties
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