This paper analyzes the daily time series of Japanese yen exchange rate against th US dollar by a model of stochastic differential equation whose coefficient oscillates cyclically. The expectation value and the variance of the series are derived by solving the equation. It is shown that their power spectrums are expressed by the power function, ƒç¡VƒØ, where ƒç is the angular frequency and 1.9 < ƒØ < 2.6. It is also revealed that the power spectrum of the expectation value has two breaking points and that of the variance has one breaking point.

The phenomenon of the power law is theoretically a puzzle. Why is the power law observed in many fields? What is the mechanism of the law? A text book by Mantegna and Stanley (2000) gives a survey of these problems in financial data.

In this paper, we take the daily time series of Japanese yen exchange rate against the US dollar and try to answer some of the above mentioned questions. Next, we shall propose a model of Stochastic Differential Equation (SDE) which is expected to reproduce the power law behavior of the expectation value ) ( ~ t x and the variance v ( t) of x ( t). If the SDE is reducible, we can get ) ( ~ t x and v ( t) as time series. Therefore, we do not have to use the method of the time window to get the variance. We shall discretize the equation and estimate the parameters using the Generalized Method of Moments (GMM). Then, we shall solve the SDE analytically and derive ) ( ~ t x and v ( t). Next, the power spectrums S ( ) of ) ( ~ t x and v ( t) are derived and fitted to power functions by the method of nonlinear regression. It will be shown that S ( ) for ) ( ~ t x has two breaking points and for v ( t) has one breaking point. Finally, we make some comments on the results obtained.

Power Laws in Exchange Rate Fluctuations, Japanese yen exchange rate, stochastic differential equation, SDE, power spectrums, angular frequency, financial data, power functions, Generalized Method of Moments, GMM, economic time series, Ornstein-Uhlenbeck equation.