A process very similar to multifractional Brownian motion

In Ayache and Taqqu (2005), the multifractional Brownian (mBm) motion is obtained by replacing the constant parameter $H$ of the fractional Brownian motion (fBm) by a smooth enough functional parameter $H(.)$ depending on the time $t$. Here, we consider the process $Z$ obtained by replacing in the wavelet expansion of the fBm the index $H$ by a function $H(.)$ depending on the dyadic point $k/2^j$. This process was introduced in Benassi et al (2000) to model fBm with piece-wise constant Hurst index and continuous paths. In this work, we investigate the case where the functional parameter satisfies an uniform H"older condition of order $\beta>\sup_{t\in \rit} H(t)$ and ones shows that, in this case, the process $Z$ is very similar to the mBm in the following senses: i) the difference between $Z$ and a mBm satisfies an uniform H"older condition of order $d>\sup_{t\in \R} H(t)$; ii) as a by product, one deduces that at each point $t\in \R$ the pointwise H"older exponent of $Z$ is $H(t)$ and that $Z$ is tangent to a fBm with Hurst parameter $H(t)$.

💡 Research Summary

The paper investigates a stochastic process Z that is constructed by modifying the wavelet series representation of fractional Brownian motion (fBm). In the classical fBm expansion, each wavelet coefficient is multiplied by a factor 2^{-j(H+1/2)} where H is the constant Hurst exponent. In multifractional Brownian motion (mBm) the constant H is replaced by a smooth time‑dependent function H(t), which yields a process whose local regularity varies with time.
Instead of letting the Hurst parameter depend on the continuous time variable, the authors replace H by a function H(k/2^{j}) that depends on the dyadic location k/2^{j} of each wavelet coefficient. The resulting process is
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