On local path behavior of Surgailis multifractional processes

International audienceMultifractional 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 Motions (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 then 10 years later, using a different construction method, which basically relies 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. Thus it turns out 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