Hideki Takayasu

Statistical properties of deterministic threshold elements — the case of market price

We analyze statistical properties of a set of deterministic threshold elements which is introduced as a model for the stock market. The macroscopic variable of the stock market price shows seemingly stochastic fluctuation with a f-2 power spectrum consistent with real economic fluctuations. The maximum Lyapunov exponent is estimated to be zero indicating that the system is at the edge of chaos.

Analysis of high-resolution foreign exchange data of USD-JPY for 13 years

We analyze high-resolution foreign exchange data consisting of 20 million data points of USD-JPY for 13 years to report firm statistical laws in distributions and correlations of exchange rate fluctuations. A conditional probability density analysis clearly shows the existence of trend-following movements at time scale of 8-ticks, about 1 minute.

Predictability of currency market exchange

We analyze tick data of yen–dollar exchange with a focus on its up and down movement. We show that there exists a rather particular conditional probability structure with such high frequency data. This result provides us with evidence to question one of the basic assumptions of the traditional market theory, where such bias in high frequency price movements is regarded as not present. We also construct systematically a random walk model reflecting this probability structure.

Statistical properties of aggregation with injection

We study a generalized aggregation process in which charged particles diffuse and coalesce randomly on a lattice. For one-dimensional and mean-field models, we show that there exists a statistically-invariant steady state when randomly charged particles are continuously injected. The steady-state charge distribution obeys a power law with the exponent depending both on the type of the injection and on the spatial dimension. The response of the system to a perturbation (i.e., relaxation) is characterized by either a power law decay (t−β,β⩽1) or a compressed exponential decay [exp(−tα),α>1].

Fractal features of the earthquake phenomenon and a simple mechanical model

A simplified two-dimensional stick-slip model is introduced. The system shows a kind of dynamical order-disorder phase transition with a control parameter proportional to the strength of interaction. At the critical point, this model is consistent with the empirical laws of earthquakes such as Gutenberg-Richters law, the spatial distribution of hypocenters, and the correlation function of the occurrence of earthquakes as results of a dynamical critical phenomenon. This model can naturally explain the locality that local magnitude distributions often deviate from Gutenberg-Richters law.

Dynamical phase transition in threshold elements

Abstract Statistical properties of threshold elements are analyzed for a model of an earthquake. The system shows a kind of percolation phase transition via mutual entrainment. The dynamical effect of cluster-cluster entrainment considerably modifies the critical exponent for cluster size distribution.

Application of the coherent anomaly method to percolation

Applying the coherent anomaly method (CAM) to site percolation problems, we estimate the percolation threshold pc and critical exponents. We obtain pc=0.589, β=0.140, γ=2.426 on the two-dimensional square lattice. These values are in good agreement with the values already known. We also investigate higher-dimensional cases by this method.

The mechanism of double exponential growth in hyper-inflation

Analyzing historical data of price indices, we find an extraordinary growth phenomenon in several examples of hyper-inflation in which, price changes are approximated nicely by double-exponential functions of time. In order to explain such behavior we introduce the general coarse-graining technique in physics, the Monte Carlo renormalization group method, to the price dynamics. Starting from a microscopic stochastic equation describing dealers’ actions in open markets, we obtain a macroscopic noiseless equation of price consistent with the observation. The effect of auto-catalytic shortening of characteristic time caused by mob psychology is shown to be responsible for the double-exponential behavior.

THE BEHAVIOR OF A THRESHOLD MODEL OF MARKET PRICE IN STOCK EXCHANGE

We analyze the behavior of deterministic threshold dynamics in a model of stock market. We observe global trends in the virtual market prices and find a kind of phase transition. At the critical region, the macroscopic variable of stock market price shows seemingly stochastic fluctuation with f-2 power spectrum consistent with real economic fluctuations. The maximum Lyapunov exponent is estimated to be slightly positive in short time steps (5 or 10 steps) and, as the observation time becomes longer, it converges to zero. This result indicates that the system is at the edge of chaos.

Self-modulation processes and resulting generic 1/f fluctuations

We analyze high precision data of transaction intervals in a foreign exchange market, and show that it is nicely approximated by a non-stationary Poisson process whose expectation value is given by a moving average of its own trace. Generalizing this result we introduce novel stochastic processes called the self-modulation processes. By the self-modulation effect, clustering occurs automatically resulting in fat-tailed interval distributions including the Zipfs law in an extreme case. We prove rigorously that the corresponding power spectrum follows the 1/f spectrum.