Fractional derivatives of random walks: Time series with long-time memory
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 models with conditional heteroskedasticity (FIGARCH), 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.
- Publication:
-
Physical Review E
- Pub Date:
- September 2008
- DOI:
- 10.1103/PhysRevE.78.031127
- arXiv:
- arXiv:0806.3171
- Bibcode:
- 2008PhRvE..78c1127R
- Keywords:
-
- 05.40.-a;
- 89.65.Gh;
- 89.75.-k;
- Fluctuation phenomena random processes noise and Brownian motion;
- Economics;
- econophysics financial markets business and management;
- Complex systems;
- Condensed Matter - Statistical Mechanics;
- Quantitative Finance - Statistical Finance
- E-Print:
- 10 pages, 14 figures