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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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