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Forma, Vol. 31 (Special Issue), pp. S41S53, 2016
doi:10.5047/forma.2016.s006

Phase Transition with Noise for Time Series of JP/US$ Exchange Rate

Hiroki Takada

Graduate School of Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui 910-8507, Japan
E-mail address: takada@u-fukui.ac.jp

(Received May 25, 2015; Accepted May 12, 2016)

Abstract. Any factor that exert the influence on the process of JP/US$ exchanges rate has been discussed widely by researchers in academic and business fields. In some cases, for instance, the cause of changes was explained by many macro-economical factors and they tried to explain by stochastic and statistical model. However, a heavy fall was necessarily expressed. We attempt to express the change of the JP/US$ exchange rate in addition to it by a phase transition model with noise. We carried out some empirical studies with a fractal analysis and statistical tests for verifying whether the series are random and generated by the stochastic model. We found that time series on difference of JP/US$ exchange rates satisfy these properties. We composed a stochastic differential equation as the model that might describe variability of the difference. We regressed the potential function in the sense of the time average on the stochastic differential equation with use of graphs of polynomials. This paper also presents the necessity of the number of order on the polynomials with consideration on stability of singular points for perturbation, what we call structural stability in Topology. With integration of the series reproduced by our model, we obtained series as numerical simulations of the JP/US$ exchange rate. We compare these series with the time series data of the JP/US$ exchange rate to evaluate our stochastic differential equation.

Keywords: Central Rate, Stochastic Differential Equation (SDE), Stability on the Form of Probability Density Function


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