Stochastic process with random increments from a symmetric stable distribution with α = 1.7. Notice the discontinuous changes.Stochastic process with random increments from a standard normal distribution.
The seven states of randomness in probability theory, fractals and risk analysis are extensions of the concept of randomness as modeled by the normal distribution. These seven states were first introduced by Benoît Mandelbrot in his 1997 book Fractals and Scaling in Finance, which applied fractal analysis to the study of risk and randomness.[1] 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 is that methods such as Markowitz mean variance portfolio and Black–Scholes model may be invalidated as the tails of the distribution of returns are fattened: the former relies on finite standard deviation (volatility) and stability of correlation, while the latter is constructed upon Brownian motion.
^Mandelbrot, Benoit B. (1997-09-18). Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E. Springer New York. ISBN 978-0-387-98363-9.
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