Fractional derivatives of random walks: Time series with long-time memory

TL;DR

This paper explores how applying fractional derivatives to random walks creates models with long-term memory, compares them to existing econometric models, and evaluates their effectiveness in simulating financial time series.

Contribution

It introduces a novel approach of using fractional derivatives on random walks to model long-memory processes and compares this with established FIGARCH models.

Findings

Fractional derivatives induce slowly-decaying autocorrelations.

Correlated random walks can effectively simulate financial time series.

Comparison shows similarities and differences with FIGARCH models.

Abstract

We review statistical properties of models generated by the application of a (positive and negative order) fractional derivative operator to a standard random walk and show that the resulting stochastic walks display slowly-decaying autocorrelation functions. The relation between these correlated walks and the well-known fractionally integrated autoregressive (FIGARCH) models, commonly used in econometric studies, is discussed. The application of correlated random walks to simulate empirical financial times series is considered and compared with the predictions from FIGARCH and the simpler FIARCH processes. A comparison with empirical data is performed.