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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 α-stable motion, defined by

Y(t)= Ω ' l(x,t)(ω ' )M(dω ' ,dx),

where l(x,t)(ω ' ) is the jointly continuous local time of a fractional Brownian motion with self-similarity index H(0,1) defined on a probability space (Ω ' , ' , ' ), and where M is an SαS random measure on the space Ω ' × with control measure ' ×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/α 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|) 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.

60G18Self-similar processes
60G52Stable processes
60G17Sample path properties