Characteristics of Critical Fluctuations Caused by Self-Modulation and its Application

我分析了所谓的自调制系统，这是一个动态随机过程，其参数由其自身轨迹的移动平均值给出。在这三年中，我研究了自调制系统的基本性质和应用，最典型的例子是在公开市场上的交易间隔，如日元-美元交易市场。我已经证明了由最优移动平均归一化的时间序列可以很好地近似于纯泊松过程。这里，最优移动平均线由特征时间为30秒的指数衰减的加权移动平均线给出，然后将自调制的基本思想推广到市场价格，特别是我拥有10多年高质量数据的日元-美元市场。我首先为市场价格引入了一个最优移动平均线，然后观察了这个噪声降低的市场价格的动态。研究发现，动态并不像金融技术所假设的那样是纯粹随机的，我发现了明确的证据，表明存在一种潜在的市场力量。这个势力不是一个常数，但它的中心是以一个更大尺度的移动平均线移动的，势的曲率也是在一个正的稳定状态到一个负的不稳定状态之间缓慢变化的。这种动力学性质可以用势系数来表征。由于这种分析方法适用于市场风险评估，本文提出了一种适用于真实的公开市场的真实的时间数据分析算法，并从理论上证明了势力公式表示为一种价差自调制过程，从而大大扩大了自调制的适用范围。

I analyzed so-called the self-modulation systems which are dynamical stochastic processes whose parameters are given by the moving averages of their own traces. In these 3 years I have studied the basic properties and also applications.The most typical example of the self-modulation systems is the transaction intervals in open markets such as the Yen-Dollar exchange market. I have shown that the time sequence of the intervals normalized by an optimal moving average is quite well approximated by a pure Poisson process. Here, the optimal moving average is given by a weighted moving average with exponential decay of characteristic time about 30 seconds.Then the basic idea of self-modulation is generalized to the market prices especially to the Yen-Dollar market which I have the high quality data for more than 10 years. I firstly introduced an optimal moving average for the market prices and then observed the dynamics of this noise-reduced market price. It is found that the dynamics is not purely random as assumed by the financial technology, I found clear evidence that there exists a potential force of market. This potential force is not a constant but its center is moving with a larger scale moving average and the curvature of the potential is also changing slowly between a positive stable state to a negative unstable state. This dynamical property can be characterized by the potential coefficient. As this analysis is suitable for risk evaluation of markets, I developed a real time data analysis algorithm which is ready to be applied to the real open market.It is also shown theoretically that the potential force formulation is expressed as a kind of self-modulation process for the price difference, so the applicability of the self-modulation has been enlarged considerably.