  In this article we will present a new perspective on the variable order
fractional calculus, which allows for differentiation and integration to a
variable order, i.e. one differentiates (or integrates) a function along the
path of a regularity function. The concept of multifractional calculus has been
a scarcely studied topic within the field of functional analysis in the last 20
years. We develop a multifractional derivative operator which acts as the
inverse of the multifractional integral operator. This is done by solving the
Abel integral equation generalized to a multifractional order. With this new
multifractional derivative operator, we are able to analyze a variety of new
problems, both in the field of stochastic analysis and in fractional and
functional analysis, ranging from regularization properties of noise to
solutions to multifractional differential equations. In this paper, we will
focus on application of the derivative operator to the construction of strong
solutions to stochastic differential equations where the drift coefficient is
merely of linear growth, and the driving noise is given by a non-stationary
multifractional Brownian motion with a Hurst parameter as a function of time.
The Hurst functions we study will take values in a bounded subset of (0,1/2).
The application of multifractional calculus to SDE's is based on a
generalization of the works of D. Nualart and Y. Ouknine from 2002.
