Stochastic Duffing equation in modelling of financial time series

Abstract

In this study, we aim to introduce a new stochastic modelling and forecasting method for the chaotic noisy Duffing map series and then for the financial time series with similar properties. We examine the fractal properties of the noisy Duffing map by correlation dimension, wavelet analysis and MFDFA analysis. Then, it is shown that the non-linear deterministic multifractal Duffing map with additive white noise process exhibits the telegraph process modulated with the fractional Gaussian noise. The combination of these stochastic processes and the fractional Gaussian noise fit the Duffing map series with white noise which has regime switching and long-range dependent properties and then the real-world financial time series with similar properties. Numerical experiments show that the proposed model generates remarkable forecast result in different horizons when it is compared with the actual outcomes. The results put forward that the chaotic Duffing equation can be a fitting model to certain financial time series or the time series with the same characteristics.