Theory of Probability and Mathematical Statistics
On local path behavior of Surgailis multifractional processes
A. Ayache and F. Bouly
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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 of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H of the well-known Fractional Brownian Motion by a deterministic function 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 and {Y(t)}t∈ℝ.
In our article, under a rather weak condition on the functional parameter H(·), we show that {M(t)}t∈ℝ and {X(t)}t∈ℝ as well as {M(t)}t∈ℝ and {Y(t)}t∈ℝ only differ by a part which is locally more regular than {M(t)}t∈ℝ itself. On one hand this result implies that the two non-classical multifractional processes {X(t)}t∈ℝ and {Y(t)}t∈ℝ have exactly the same local path behavior as that of the classical MBM {M(t)}t∈ℝ. On the other hand it allows to obtain from discrete realizations of {X(t)}t∈ℝ and {Y(t)}t∈ℝ strongly consistent statistical estimators for values of their functional parameter.
Keywords: Gaussian processes, variable Hurst parameter, local and pointwise Hölder regularity, local self-similarity
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