Self-Excited Multifractal Dynamics

TL;DR

The paper introduces the self-excited multifractal (SEMF) model, capturing endogenous self-organization in complex systems with properties like multifractality, heavy tails, and long-range correlations, relevant to turbulence, earthquakes, and financial markets.

Contribution

The SEMF model is a novel approach combining self-excitation and exponential nonlinearity to reproduce key stylized facts of financial time series and other complex systems.

Findings

Reproduces multifractality and heavy tails in data

Captures long-range correlations of squared increments

Displays leverage effect and time-reversal asymmetry

Abstract

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.