Analysis of Stochstic Evolution

In this paper, we propose an optimization-based mechanism to explain power law distributions, where the function that the optimization process is seeking to optimize is derived mathematically, then the behavior and interpretation of this function are analyzed. The derived function shows some similarity to the entropy function in representing order and randomness; however, it also represents the energy, where the optimization process is seeking to maximize the number of elemen...

In this paper, we quantitatively investigate the statistical properties of a statistical ensemble of stock prices. We selected 1200 stocks traded on the Tokyo Stock Exchange, and formed a statistical ensemble of daily stock prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period of the forming of the internet bubble in Japn, and its bursting in the Japanese stock market. We found that the tail of the complementar...

We use the formalism of 'Maximum Principle of Shannon's Entropy' to derive the general power law distribution function, using what seems to be a reasonable physical assumption, namely, the demand of a constant mean "internal order" (Boltzmann Entropy) of a complex, self interacting, self organized system. Since the Shannon entropy is equivalent to the Boltzmann's entropy under equilibrium, non interacting conditions, we interpret this result as the complex system making use o...

Real world markets display power-law features in variables such as price fluctuations in stocks. To further understand market behavior, we have conducted a series of market experiments on our web-based prediction market platform which allows us to reconstruct transaction networks among traders. From these networks, we are able to record the degree of a trader, the size of a community of traders, the transaction time interval among traders and other variables that are of int...

This paper considers a general one-dimensional stochastic differential equation (SDE). A particular attention is given to the SDEs that may be transformed (via Ito's formula) into:$$d X\_t = ( \bar{B} (X\_t) - b X\_t) d t + \sqrt{X\_t} d W\_t, ~~~X\_0 > 0,$$where $ \bar{B}(y)/ y \to 0$. It is shown that the MGF of $X\_t$ explodes at a critical moment $\mu^\ast\_t$ which is independent of $\bar{B}$. Furthermore, this MGF is given as a sum of the MGF of a Cox-Ingersoll-Ross pro...

We study the rank distribution, the cumulative probability, and the probability density of returns of stock prices of listed firms traded in four stock markets. We find that the rank distribution and the cumulative probability of stock prices traded in are consistent approximately with the Zipf's law or a power law. It is also obtained that the probability density of normalized returns for listed stocks almost has the form of the exponential function. Our results are compared...

Power-law distributions have been widely observed in different areas of scientific research. Practical estimation issues include how to select a threshold above which observations follow a power-law distribution and then how to estimate the power-law tail index. A minimum distance selection procedure (MDSP) is proposed in Clauset et al. (2009) and has been widely adopted in practice, especially in the analyses of social networks. However, theoretical justifications for this s...

We present a new method of estimating the distribution of sales rates of, e.g., book titles at an online bookstore, from the time evolution of ranking data found at websites of the store. The method is based on new mathematical results on an infinite particle limit of the stochastic ranking process, and is suitable for quantitative studies of the long tail structure of online retails. We give an example of a fit to the actual data obtained from Amazon.co.jp, which gives the P...

Probability distributions having power-law tails are observed in a broad range of social, economic, and biological systems. We describe here a potentially useful common framework. We derive distribution functions $\{p_k\}$ for situations in which a `joiner particle' $k$ pays some form of price to enter a `community' of size $k-1$, where costs are subject to economies-of-scale (EOS). Maximizing the Boltzmann-Gibbs-Shannon entropy subject to this energy-like constraint predicts...

We show that recent stock market fluctuations are characterized by the cumulative distributions whose tails on short, minute time scales exhibit power scaling with the scaling index alpha > 3 and this index tends to increase quickly with decreasing sampling frequency. Our study is based on high-frequency recordings of the S&P500, DAX and WIG20 indices over the interval May 2004 - May 2006. Our findings suggest that dynamics of the contemporary market may differ from the one o...