Ornstein-Uhlenbeck Process via Conflated Drive of Brownian Motion and Lévy Process and its Application

Abstract

Non-linear time series and linear models were not designed to detect probabilistic process that are depict by velocity and drift associated to returns the way Ornstein-Uhlenbeck stochastic process describes diffusion and velocity associated to series or waves influenced by Brownian motion or Lévy process.  In this research, Brownian motion and Lévy process were conflated as driving force for Ornstein-Uhlenbeck process with its solution applied to Naira-Dollar exchange rates from 2009-2019.The drift and diffusion estimates for the Ornstein-Uhlenbeck process driven by Brownian motion and Lévy process are realization of AR (1) with 2.991 and 0.1672 respectively. The AR(1) realization for the Ornstein-Uhlenbeck process was stationary with estimate  that lies outside the unit circle. The AIC, BIC, RMSE, and MSE for the Ornstein-Uhlenbeck process were estimated to be 483.7572, 483.4782, 0.00101, and 8.395 respectively, compare to estimates of the same indexes for AR (1) of 767.5, 634.09, 0.3819, and 23.48. The criterion via the residuals from the Ornstein-Uhlenbeck process was smaller, which connotes that the errors approximated in using drift, Brownian motion and to estimate  is relatively small via the Ornstein-Uhlenbeck process.

Keywords: Brownian motion; Drift; Diffusion; Lévy process; Ornstein-Uhlenbeck Process

DOI: 10.7176/MTM/11-3-02

Publication date:May 31^st 2021