  We introduce the self-excited multifractal (SEMF) model, defined such that
the amplitudes of the increments of the process are expressed as exponentials
of a long memory of past increments. The principal novel feature of the model
lies in the self-excitation mechanism combined with exponential nonlinearity,
i.e. the explicit dependence of future values of the process on past ones. The
self- excitation captures the microscopic origin of the emergent endogenous
self-organization properties, such as the energy cascade in turbulent flows,
the triggering of aftershocks by previous earthquakes and the "reflexive"
interactions of financial markets. The SEMF process has all the standard
stylized facts found in financial time series, which are robust to the
specification of the parameters and the shape of the memory kernel:
multifractality, heavy tails of the distribution of increments with
intermediate asymptotics, zero correlation of the signed increments and
long-range correlation of the squared increments, the asymmetry (called
"leverage" effect) of the correlation between increments and absolute value of
the increments and statistical asymmetry under time reversal.
