Time series obtained by solving non-linear stochastic models exhibit rather interesting statistical properties. On Physics of Risk we have already discussed some of these models [1, 2] (ex. stochastic model of return, herding model of financial markets), which are able to reproduce statistical properties of high frequency return (namely spectral density and probability distribution).

In statistical sense model and financial market behavior might be studied in many different manners. One may study probability distributions, moments, spectral densities, autocorrelations and etc., using each of them to obtain vital information on the statistical and dynamical properties of the studied system. It is important to note that new useful information might be provided by the statistical indicators, which are related to the previously used indicators in unambiguous manner. One may also introduce new variables describing system itself or its time series.

There is a group of such variables, which is closely related to the estimation of risk, known as burst statistics [3, 4]. In this text we will discuss these variables and their statistical properties. At the end of the text we also present an interactive HTML5 applet, using which one can reproduce burst statistics of certain stochastic model.

Kirman's ant colony model, previously presented on our website as agent based (based on [1]) and stochastic (based on [2, 3]) model, has become classical example of herding modeling. Application of this model towards economic, financial or other social scenarios might seem doubtful as human society is far more complex than ant colony, but methodologically it is more useful to start from very simple and stylized model and later add complexity on top of it. Furthermore we have already shown that Kirman's herding dynamics could be applicable in agent-based marketing (see comparison of Kirman's and Bass diffusion model). In this text we will consider financial market scenario and obtain stochastic differential equations similar to the existing stochastic models considered in [4, 5].

Working on Physics of Risk is very interesting and useful experience. This experience provides valuable insights into the mechanics behind various complex systems, well modeled by macroscopic models. Using our experience we are able to obtain qualitative and quantitative agreements between varying models. In our newest publication [1] we have used one-step formalism [2] to obtain macroscopic treatments of Kirman model [3].

Bass diffusion model [1] is widely known and very important model in marketing science. This model predicts diffusion, sales, of new successful products inside the market. While previously discussed model, Kirman model [2], has wide range of possible applications - from biology to finance. In [2] it is noted that behavior observed by entomologists is similar to one observed in economic scenarios (ex. popularity of books and restaurants). Thus one might expect that Kirman's model might be modified to work in the marketing scenario.

Newton-Raphson, sometimes just Newton or Newton-Fourier, method is an approximate method in mathematical analysis for finding local roots of very complex functions (such as polynomials with large powers). Recall that root of the function is defined as a solution of \( f(z) = 0 \). The essence of this method is to linearize function at the guessing point. The point where linearized function passes the abscissa axis is assumed to be a more precise estimate of the actual root. Mathematically: