Jun-ichi Maskawa

Theory and numerical calculation of pattern formation in shrinking gels

We discuss pattern formation in three-dimensional gels with cylindrical shapes during their shrinking such as volume phase transition. A Ginzburg–Landau theory is given for the pattern formation in shrinking gels. A characteristic feature in shrinking gels is the dense layer formed around the gel surface in the early stage of phase transition. This layer reduce considerably permeation of solvent and the shrinkage practically stops. We introduce the external osmotic pressure and the external elastic stress acting on the gel surface in order to take account of the effect of the layer. Patterns are classified according to the anisotropy and the incompressibility of gels by the linearized stability analysis of the theory. It appears that the external stress term suppresses the growth of the fluctuation with short wavelength. The results obtained by a numerical calculation for the evolution of patterns are also shown.

Stock price fluctuations and the mimetic behaviors of traders

We give a stochastic microscopic modelling of stock markets driven by continuous double auction. If we take into account the mimetic behavior of traders, when they place limit order, our virtual market shows the power-law tail of the distribution of returns with the exponent outside the Levy stable region, the short memory of returns and the long memory of volatilities. The Hurst exponent of our model is asymptotically 12. An explanation is also given for the profile of the autocorrelation function, which is responsible for the value of the Hurst exponent.

Correlation of coming limit price with order book in stock markets

We examine the correlation of the limit price with the order book, when a limit order comes. We analyzed the Rebuild Order Book of Stock Exchange Electronic Trading Service, which is the centralized order book market of London Stock Exchange. As a result, the limit price is broadly distributed around the best price according to a power-law, and it is not randomly drawn from the distribution, but has a strong correlation with the size of cumulative unexecuted limit orders on the price. It was also found that the limit price, on the coarse-grained price scale, tends to gather around the price which has a large size of cumulative unexecuted limit orders.

Multivariate Markov chain modeling for stock markets

We study a multivariate Markov chain model as a stochastic model of the price changes of portfolios in the framework of the mean field approximation. The time series of price changes are coded into the sequences of up and down spins according to their signs. We start with the discussion for small portfolios consisting of two stock issues. The generalization of our model to arbitrary size of portfolio is constructed by a recurrence relation. The resultant form of the joint probability of the stationary state coincides with Gibbs measure assigned to each configuration of spin glass model. Through the analysis of actual portfolios, it has been shown that the synchronization of the direction of the price changes is well described by the model.

Ordered phase and non-equilibrium fluctuation in stock market

We analyze the statistics of daily price change of stock market in the framework of a statistical physics model for the collective fluctuation of stock portfolio. In this model the time series of price changes are coded into the sequences of up and down spins, and the Hamiltonian of the system is expressed by spin–spin interactions as in spin glass models of disordered magnetic systems. Through the analysis of Dow–Jones industrial portfolio consisting of 30 stock issues by this model, we find a non-equilibrium fluctuation mode on the point slightly below the boundary between ordered and disordered phases. The remaining 29 modes are still in disordered phase and well described by Gibbs distribution. The variance of the fluctuation is outlined by the theoretical curve and peculiarly large in the non-equilibrium mode compared with those in the other modes remaining in ordinary phase.

Stock price process and long memory in trade signs

Empirical study on tick by tick data in stock markets shows us that there exists a long memory in trade signs and signed trade volumes. This means that an order flow is a highly autocorrelated long memory process. We present a mathematical model of trade signs and trade volumes in which traders decompose their orders into small pieces. We prove that fractional Brownian motions are obtained as a scaling limit of the signed volume process induced by the model.

Gibbs measure and Markov chain modeling for stock markets

We reviewed the recent work on Gibbs measure (statistical physics model) describing the collective price jumps in stock markets. We started with the study of a multivariate Markov chain model as a. stochastic model of the price changes of portfolios in the framework of the mean field approximation. The time series of price changes were coded into the sequences of up and down spins according to their signs. As the stationary state of the Markov chain, Gibbs measure was naturally derived, which formally coincides with spin glass model of disordered magnetic systems. The linear response of the system to external fields was examined to prove the fluctuation response theorem, Finally, the analysis of actual portfolios based on this model was briefly summarized.

Collective Behavior of Market Participants during Abrupt Stock Price Changes.

Under uncertainty, human and animal collectives often respond stochastically to events they encounter. Human or animal individuals behave depending on others’ actions, and sometimes follow choices that are sub-optimal for individuals. Such mimetic behaviors are enhanced during emergencies, creating collective behavior of a group. A stock market that is about to crash, as markets did immediately after the Lehman Brothers bankruptcy, provides illustrative examples of such behaviors. We provide empirical evidence proving the existence of collective behavior among stock market participants in emergent situations. We investigated the resolution of extreme supply-and-demand order imbalances by increased balancing counter orders: buy and sell orders for excess supply and demand respectively, during times of price adjustment, so-called special quotes on the Tokyo Stock Exchange. Counter orders increase positively depending on the quantity of revealed counter orders: the accumulated orders in the book until then. Statistics of the coming counter order are well described using a logistic regression model with the ratio of revealed orders until then to the finally revealed orders as the explanatory variable. Results given here show that the market participants make Bayesian estimations of optimal choices to ascertain whether to order using information about orders of other participants.

Spin-glass like network model for stock market

A statistical physics model for the collective price changes of stock portfolios is proposed. That is an analogue to spin glass model (Mezard et al. 1987) for disordered magnetic system. In this model the time series of price changes are coded into the sequences of up and down spins. The Hamiltonian of the system is expressed by long-range spin-spin interactions as in SherringtonKirtpatrick (S-K) model (Sherrington et al. 1975) of spin glass. The interaction coefficients between two stocks are determined by empirical using fluctuationresponse theorem.

Statistical physics model for the collective price fluctuations of portfolios

A statistical physics model for the collective price changes of stock portfolios is propose; it is an analogue to the spin glass model for a disordered magnetic system. In this model the time series of price changes are coded into the sequences of up and down spins. The Hamiltonian of the system is expressed by long-range spin-spin interactions as in the Sherrington-Kirkpatrick model of spin glass (D. Sherrington and S. Kirkpatrick, 1975). The interaction coefficients between two stocks are determined by empirical data using fluctuation-response theorem. Our theory is applied to price changes of stocks in the Dow-Jones industrial portfolio. Monte Carlo simulations are performed based on the model. The resultant probability distributions of magnetization show good agreement with empirical data.