  The Ornstein-Uhlenbeck process can be seen as a paradigm of a finite-variance
and statistically stationary rough random walk. Furthermore, it is defined as
the unique solution of a Markovian stochastic dynamics and shares the same
local regularity as the one of the Brownian motion. A natural generalization of
this process able to reproduce the local regularity of a fractional Brownian
motion of parameter H is provided by the fractional Ornstein-Uhlenbeck process.
Based on previous works, we propose to include some Multifractal corrections to
this picture using a Gaussian Multiplicative Chaos. The aforementioned process,
called a Multifractal fractional Ornstein-Uhlenbeck process, is a statistically
stationary finite-variance process. Its underlying dynamics is non-Markovian,
although non-anticipating and causal. The numerical scheme and theoretical
approach are based on a regularization procedure, that gives a meaning to this
dynamical evolution, which unique solution converges towards a well-behaved
stochastic process.
