Levy walk and Fluctuations in Finance. Change in asset price : behaves like a random walk (PhD thesis, L. Bachelier, 1900). It’s a Levy distribution ! Mandelbrot, 1963. Levy distribution has fat tails ! ( Sometimes markets do crash !! )
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Change in asset price :
behaves like a random walk (PhD thesis, L. Bachelier, 1900).
It’s a Levy distribution !
( Sometimes markets do crash !! )
Self similar with respect to sampling time windowT .
(ie, whether data is collected every
five minutes or every month).
Log PDF vs logX
ie, correlation time very short
(Hence Markovian , like random walk ! )
Otherwise you could predict the market
and easily become millionaire.
Mass Transport and DiffusionRandom Walk Diffusion( a slow process <x2> = 2Dt )Diffusion can be enhanced or slowed down by flows or potentials. Tracers in turbulent flows. <x2> ~ t 3 (Richardson’s law)
drawing: Leonardo da Vinci,1500
(a non-equilibrium problem)
The famous Kramers formula (1940)
Application: rate of chemical reactions, lifetime of weak bonds
Transport inside a cell (size ~ 10 m)
Random walkers transport cargo through highways !!
The cell has network of filaments (microtubules) along which the
the motor proteins (random walkers) move with their cargo.
But unbiased random
walk cannot generate
Need help from
V(x,t) switches between V1 and V2