A Process Very Similar to Multifractional Brownian Motion

Summary

Multifractional Brownian motion (mBm), denoted here by X, is one of the paradigmatic examples of a continuous Gaussian process whose pointwise Hölder exponent depends on the location. Recall that X can be obtained (see, e.g., Ayache and Taqqu Publ Mat 49:459–486,2005; Benassi et al. Rev Mat Iberoam 13:19–81, 1997) by replacing the constant Hurst parameter H in the standard wavelet series representation of fractional Brownian motion (fBm) by a smooth function H( ⋅) depending on the time variable t. Another natural idea (see Benassi et al. Stat Infer Stoch Proc 3:101–111, 2000) which allows us to construct a continuous Gaussian process, denoted by Z, whose pointwise Hölder exponent does not remain constant all along its trajectory, consists in substituting H(k ∕ 2^j) to H in each term of index (j, k) of the standard wavelet series representation of fBm. The main goal of our chapter is to show that, under some assumption on the bounds of H( ⋅), X and Z only differ by a process R which is smoother than they are; this means that they are very similar from a fractal geometry point of view.