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) {M(t)}(t is an element of R) of Benassi, Jaffard, Levy Vehel, Peltier and Roux, which was constructed in the mid 90's just by replacing the constant Hurst parameter g-C of the well-known Fractional Brownian Motion by a deterministic function g-C(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 is an element of R), and {Y(t)}(t is an element of R). In our article, under a rather weak condition on the functional parameter H(.), we show that {M(t)}(t is an element of R) and {X(t)}(t is an element of R) as well as (M(t)}(t is an element of R) and {Y(t)}(t)(is an element of R) only differ by a part which is locally more regular than {M(t)}(t is an element of R) itself. On one hand this result implies that the two non-classical multifractional processes {X(t)}(t is an element of R) and {Y(t)}(t is an element of R) have exactly the same local path behavior as that of the classical MBM {NtiltER. On the other hand it allows to obtain from discrete realizations of {X(t)}(t is an element of R) and {Y(t)}(t is an element of R) strongly consistent statistical estimators for values of their functional parameter.