Narrow Results

The problem of escape of a Brownian particle in a cusp-shaped metastable potential is of special importance in nonadiabatic and weakly-adiabatic rate theory for electron transfer (ET) reactions. For the weakly-adiabatic reactions, the reaction follows an adiabaticity criterion in the presence of a sharp barrier. In contrast to the nonadiabatic case, the ET kinetics can be, however considerably influenced by the medium dynamics. In this paper, the problem of the escape time over a dichotomously fluctuating cusp barrier is discussed with its relevance to the high temperature ET reactions in condensed media.

We have modified the transition probability calculations in the lattice gas model of biased-random walkers. Adopting the collision-avoided rules, a new cellular automata model of pedestrian flow which updates in parallel is established. Using this model, we have simulated and studied the evacuation process of people in a hall. The spatial configuration of people is presented, which is more consistent with the scene of reality than that presented in the lattice-gas model. It is found that the escape time t_c closely depends on the width of the door and the initial density of people, it scales as t_c ∝ W^-0.55±0.01 and t_c ∝ ρ^0.98±0.01 ≈ ρ.

The average avalanche size can be calculated exactly in a number of models of self-organized criticality (SOC). While the calculation is straight-forward in one dimension, it is more involved in higher dimensions and further complicated by the presence of different boundary conditions and different forms of external driving. Amplitudes of the leading order are determined analytically and evaluated to obtain analytical references for numerical work. A subtle link exists between the procedure to calculate the average avalanche size and the field theory of SOC.

The stability of portfolio investment in stock market crashes with Markowitz portfolio is investigated by the method of theoretical and empirical simulation. From numerical simulation of the mean escape time (MET), we conclude that: (i) The increasing number $(Np)$ of stocks in Markowitz portfolio induces a maximum in the curve of MET versus the initial position; (ii) A critical value of $Np$ in the behavior of MET versus the long-run variance or amplitude of volatility fluctuations maximumlly enhances the stability of portfolio investment. When $Np$ takes value below the critical value, the increasing $Np$ enhances the stability of portfolio investment, but restrains it when $Np$ takes value above the critical value. In addition, a good agreement of both the MET and probability density functions of returns is found between real data and theoretical results.

The roles of the trading time risks (TTRs) on stock investment return and risks are investigated in the condition of stock price crashes with Hushen300 data (CSI300) and Dow Jones Industrial Average (ˆDJI), respectively. In order to describe the TTR, we employ the escape time that the stock price drops from the maximum to minimum value in a data window length (DWL). After theoretical and empirical research on probability density function of return, the results in both ˆDJI and CSI300 indicate that: (i) As increasing DWL, the expectation of returns and its stability are weakened. (ii) An optimal TTR is related to a maximum return and minimum risk of stock investment in stock price crashes.

We study the statistical properties of escape times for stock price returns in the Wall Street market. In particular we get the escape time distribution for real data from daily transactions and for three models: (i) the Wiener process with drift and a constant market volatility, (ii) Heston and (iii) GARCH models, where the volatility is a stochastic process. We find that the first model is unable to catch all the features of the escape time distribution of real data. Moreover, the Heston model describes the probability density function for both return and escape times better than the GARCH model.