Modelling Nonlinear Dynamical Systems Using Itô Stochastic Differential Equations With Jump Process

Diffusion processes have been used for describing nonlinear dynamical systems and the geometric Brownian motion has long served as a foundational model for capturing stochastic nature of processes characterized by the continuous random fluctuations. This study developed a modification to Lévy process for the Nigeria exchange rate to the US dollar using Ito stochastic differential equation by considering the case where the price movement involves sudden jumps, while capturing statistical features present in the time series. The developed extended xMJNID model, assumes that the market model has no arbitrage opportunities and the exchange rate follows a Merton model. The extension was a jump composed of Poisson process with nonconstant intensity function. Through simulation study and application to real data, the xMJNID model was shown to perform better than existing diffusion models including the Merton model and ARIMA. Comparisons were made using accuracy measures and Akaike and Bayesian information criteria.