Individual
and collective learning in multi-agent Systems
1 - Markets as
cognitive and interactive complex systems
2.3. Collective learning (3) : Statistical
physics approach of markets (b)
History
From the probability theory, Bachelier (1900) first introduced a model of the stock market where
prices behave as a Brownian motion.
Levy (1937-1954, 1965) first introduce
a general theory for the sums of independent identically
distributed terms, knowed as « Stable
Distribution ». Laplace-Gauss distribution is not
the only one Stable Distribution, but some others like
Cauchy and Levy exist and have much heavier tails, which
may be turn in an infinite variance.
Mandelbrot (1959-1997) use Levy
distributions together with data from incomes and stocks
for to introduce new concepts as scale invariance,
fractality and differents forms of hazard (benign,
savage). « Savage » Hazard have two
declination : the « Noé » effect
(intermittents burst) and the « Joseph »
effect (long run dependance and fluctuations). His
numerous contributions was largely ignored by Financial
(mainly neoclassical) Economist (as Nobel Prize), but
have interested physicists.
In Economics, Pareto have proposed a
specific Distribution of Probability (the so-called Power
Law) for the distribution of incomes. Power law also
characterize distribution of the size of the firms
(Gibrat)
Real financial series are so
characterized by scalling proprieties, long range
dependance in absolute (or square) values of volatility
(increments), intermittents burst
(or crash) of all size (possibly limited by some maximum,
as "truncated levy flight"),
we turn on « heavy tailed » or
« leptokurtic » distribution
In Statistical Physics, large
fluctuations, Power Law Distribution
and scale invariance are often
signatures of an underlying critical
phenomenon. Bak introduce the concept of
« Self Organized Criticality »
in order to describe a metastable attractor characterized
by all size avalanches –
which correspond to a Power Law distribution, and long
range spatial correlation between variables.
