On local path behavior of Surgailis multifractional processes

Abstract

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