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Multifractional processes with random exponent. (English) Zbl 1082.60032
The multifractional process with random exponent (MPRE) is defined as follows. Let $B(t,H)=\int_R((t-x)_{+}^{H-1/2}-(-x)_{+}^{H-1/2})dW(x),$ where $$W$$ is the standard Wiener process, $$H\in(0,1)$$ and $$S(t)$$ is some stochastic process (maybe dependent on $$W$$ and non-stationary). Then $$Z(t)=B(t,S(t))$$ is MPRE and $$S$$ is its random exponent. (Note that for fixed $$S(t)=H$$, $$Z(t)=B(t,H)$$ is a fractional Brownian motion with Hurst parameter $$H$$.) The authors describe local properties of $$Z(t)$$, e.g. derive conditions under which the pointwise Hölder exponent of $$Z(t)$$ is $$S(t)$$ and the global Hölder exponent over $$J$$ is $$\inf_{t\in J}S(t)$$. It is shown that if $$S$$ is a stationary process independent of $$W$$, then $$Z(at)$$ has the same distribution as $$a^{S(t)}Z(t)$$ for any fixed $$a>0$$ and $$t$$. The proofs are based on wavelet decompositions of the random field $$\{B(t,H), \;t\in[0,1], H\in[a,b]\}$$.

##### MSC:
 60G18 Self-similar stochastic processes 60G17 Sample path properties
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