Question

Please elaborate on the differences and the relationships between Brownian motion, Wiener process and Levy process, and also the characteristics of each.

Thank you so much!

Answer 1

In most sources, the Brownian Motion and the Wiener Process are the same things. However, in some sources the Wiener process is the standard Brownian motion while a general Brownian Motion is of a form αW(t) + β.

A Brownian Motion or Wiener process, is both a Markov process and a martingale. These two properties are very different. In fact, they have little in common. A random process X_t, adapted to a filtration F_t, is a martingale with respect to the filtration if conditional expectation of the increment E( X_t - X_s | F_s) =0 for all t>s (conditional increment equals 0).

The same process has Markov property if

E( X_t | F_s) = E( X_t | X_s) =0 for all t>s (future distribution is independent of past).

the increments of a GBM are neither stationary nor independent.

This is the reason why GBM is not a Lévy process.