On discrete stochastic processes with long-lasting time dependence

TL;DR

This paper investigates the statistical properties of a heteroskedastic process with long-lasting memory, analyzing correlations, distributions, and multiscaling, and introduces an asymmetric variance model to replicate financial leverage effects.

Contribution

It introduces a novel heteroskedastic process with q-exponential memory and extends it to asymmetric variance to model leverage effects in finance.

Findings

Analyzed self-correlation and kurtosis of the process.

Derived stationary probability density functions.

Reproduced leverage effect with asymmetric variance model.

Abstract

In this manuscript, we analytically and numerically study statistical properties of an heteroskedastic process based on the celebrated ARCH generator of random variables whose variance is defined by a memory of $q_{m}$-exponencial, form ($e_{q_{m}=1}^{x}=e^{x}$). Specifically, we inspect the self-correlation function of squared random variables as well as the kurtosis. In addition, by numerical procedures, we infer the stationary probability density function of both of the heteroskedastic random variables and the variance, the multiscaling properties, the first-passage times distribution, and the dependence degree. Finally, we introduce an asymmetric variance version of the model that enables us to reproduce the so-called leverage effect in financial markets.