![Stable motion blue](https://upload.wikimedia.org/wikipedia/commons/f/fe/Stable_motion_blue.jpeg)
The seven states of randomness in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
,
fractals
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
and
risk analysis are extensions of the concept of
randomness
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
as modeled by the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. These seven states were first introduced by
Benoît Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
in his 1997 book ''Fractals and Scaling in Finance'', which applied
fractal analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from p ...
to the study of risk and randomness.
Benoît Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
(1997) ''Fractals and scaling in finance'' pages 136–142 https://books.google.com/books/about/Fractals_and_Scaling_in_Finance.html?id=6KGSYANlwHAC&redir_esc=y This classification builds upon the three main states of randomness: mild, slow, and wild.
The importance of seven states of randomness classification for
mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
is that methods such as
Markowitz mean variance portfolio and
Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Blac ...
may be invalidated as the tails of the distribution of returns are
fattened: the former relies on finite
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
(
volatility) and stability of
correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...
, while the latter is constructed upon
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
.
History
These seven states build on earlier work of Mandelbrot in 1963: "The variations of certain speculative prices" and "New methods in statistical economics" in which he argued that most
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
s approached only a first stage of dealing with
indeterminism
Indeterminism is the idea that events (or certain events, or events of certain types) are not caused, or do not cause deterministically.
It is the opposite of determinism and related to chance. It is highly relevant to the philosophical probl ...
in science, and that they ignored many aspects of real world
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
, in particular, most cases of
financial modeling
Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio o ...
. This was then presented by Mandelbrot in the International Congress for Logic (1964) in an address titled "The Epistemology of Chance in Certain Newer Sciences"
[B. Mandelbrot, Toward a second stage of indeterminism in Science, Interdisciplinary Science Reviews 198]
/ref>
Intuitively speaking, Mandelbrot argued that the traditional normal distribution does not properly capture empirical and "real world" distributions and there are other forms of randomness that can be used to model extreme changes in risk and randomness. He observed that randomness can become quite "wild" if the requirements regarding finite mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
and variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
are abandoned. Wild randomness corresponds to situations in which a single observation, or a particular outcome can impact the total in a very disproportionate way.
The classification was formally introduced in his 1997 book ''Fractals and Scaling in Finance'', as a way to bring insight into the three main states of randomness: mild, slow, and wild . Given ''N'' addends
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
, ''portioning'' concerns the relative contribution of the addends to their sum. By ''even'' portioning, Mandelbrot meant that the addends were of same order of magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...
, otherwise he considered the portioning to be ''concentrated''. Given the moment
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of order ''q'' of a random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, Mandelbrot called the root of degree ''q'' of such moment the ''scale factor'' (of order ''q'').
The seven states are:
# Proper mild randomness: short-run portioning is even for ''N'' = 2, e.g. the normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
# Borderline mild randomness: short-run portioning is concentrated for ''N'' = 2, but eventually becomes even as ''N'' grows, e.g. the exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
with rate ''λ'' = 1 (and so with expected value 1/''λ'' = 1)
# Slow randomness with finite delocalized moments: scale factor increases faster than ''q'' but no faster than