One of the most remarkable properties of Brownian motion

is self-similarity: for all

and all

, the random variable

has the same distribution as

. A strictly stable Lévy process

can be viewed as a generalization of Brownian motion, and has the property for all

,

,

has the same distribution as

, for some index of stability

, where

. For each index of stability

, the distributions

are stable distributions

. The probability density functions of this distribution are not known in explicit form except in special cases. In this Demonstration we consider the symmetric case, for which

. In this case the characteristic function is given by

, which shows that for

we get the Cauchy distribution and for

the normal distribution.

A symmetric

-stable process can be represented as a combination of a (compound) Poisson process and a Brownian motion. For small values of

we see that the process is dominated by big jumps. For medium values (e.g.,

, i.e., Cauchy process) we get both small and large jumps. For

close to 2 we get Brownian motion with occasional jumps.

-stable processes for

have infinite variance, which makes them somewhat inconvenient. Nevertheless, they are important in physics, biology, meteorology, and have been used in option pricing in finance.

G. Samrodnitsky and M. Taqqu,

*Stable Non-Gaussian Random Processes*, New York: Chapman and Hall, 1994.

A. N. Shiryaev,

*Essentials of Stochastic Finance*, River Edge, New Jersey: World Scientific Publishing Co., 1999.