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.