Please elaborate on the differences and the relationships between Brownian motion, Wiener process and Levy process,...

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!

Expert Solution

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.

milcah answered 1 year ago

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Please elaborate on the differences and the relationships between Brownian motion, Wiener process and Levy process,...

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