Random Walk derives from the martingale theory. The simplest definition of random walk implies that the variation of the variable is also associated with the IID (Independently and Identically Distributed) definition of the distribution of ?t.
In the probability theory, a sequence of random variables is IID only if each of these variables has a probability distribution equals to the other and mutually independent. The random walk with IID assumption of a stock price variable can be represented as:

Pt = ? + Pt-1 +?t

where ?t ?IID (o,?2)
In the previous equation, ? is the expected price variation called "drift". The presence of the drift implies that the average is non-zero. The negative or positive drift is associated with the non-stationary hypothesis. The IID (o,?2) implies that the variation of ?t is with 0 mean and ?2 (if we assume a normal distribution of ?t, the equation is an arithmetic Brownian motion observed at regular unitary intervals of regular length).
The definition of random walk given so far is the most restricted one (RW1). If we free the initial hypothesis, it is possible to compare the random walk that is closer to the financial market dynamics.
If we assume that the identically distributed variation is not plausible if we refer to the extended time length, we can define RW2. In the same way, if we only hold the uncorrelated increments we can define RW3; with this set up, we do not exclude the possibility that the square of the variation is correlated.
The main difference between RW and martingale lies in the fact that the random walk process is more restrictive than the martingale in that it requires that the value following the first (e.g. the variance) be statistically independent. Even if the martingale defines the random variation of a generic variable, it allows the possibility of foreseeing the conditioned variance on the basis of previous value.

Campbell, J. Y., Lo, W. A., MacKinlay, A. C., 1997, The Econometrics of Financial Markets, Princeton University Press, Princeton New Jersey;
Lo, W. A., MacKinlay, A. C., 1988, Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test, in The Review of Financial Studies, Vol. 1, No. 1 (Spring), pp. 41-66.


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