On local path behavior of Surgailis multifractional processes

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$ of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter ${\mathcal {H}}$ of the well-known Fractional Brownian Motion by a deterministic function ${\mathcal {H}}(t)$ having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by $\{X(t)\}_{t\in \mathbb {R}}$ and $\{Y(t)\}_{t\in \mathbb {R}}$.

In our article, under a rather weak condition on the functional parameter ${\mathcal {H}}(\cdot )$, we show that $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$ and $\{X(t)\}_{t\in \mathbb {R}}$ as well as $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$ and $\{Y(t)\}_{t\in \mathbb {R}}$ only differ by a part which is locally more regular than $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$ itself. On one hand this result implies that the two non-classical multifractional processes $\{X(t)\}_{t\in \mathbb {R}}$ and $\{Y(t)\}_{t\in \mathbb {R}}$ have exactly the same local path behavior as that of the classical MBM $\{{\mathcal {M}}(t)\}_{t\in \mathbb {R}}$. On the other hand it allows to obtain from discrete realizations of $\{X(t)\}_{t\in \mathbb {R}}$ and $\{Y(t)\}_{t\in \mathbb {R}}$ strongly consistent statistical estimators for values of their functional parameter.