Random Walk
 Explained:

A random walk is a simple type of discrete stochastic process whose increments form a white noise. Since a white noise has zero mean, a random walk is a martingale. Let's formalize this.

If you have not already done so, see the notation conventions documentation. A discrete univariate stochastic process R is called a random walk if its increments

 tW = tR – t–1R [1]

form a white noise. Because there are different types of white noises, there are different types of random walks. A simple random walk—what probability theorists generally call a random walk—is one whose increments form a strong white noise whose terms only take on the values 1 or –1, each with probability 0.5. A realization of a simple white noise is indicated in Exhibit 1 along with the corresponding realization of the white noise of its increments.

 Simple Random Walk Exhibit 1 The top graph indicates a realization of a simple white noise. The bottom graph indicates the corresponding realization of the white noise of its increments.

The top graph of Exhibit 1 illustrates how a simple random walk takes random "steps" up or down, which is what motivated the name "random walk."

In finance, an arithmetic random walk is a random walk with increments that are a Gaussian white noise. This can be represented as

 tR – t–1R = tN [2]

where the tN are independent and identically distributed standard normal random variables, and is constant. If a constant drift term is added, this becomes an arithmetic random walk with drift:

 tR – t–1R = + tN [3]

For modeling the behavior of certain asset prices, such as stock prices, arithmetic random walks have a number of limitations. They can take on negative values, which is an impossibility for many asset's prices. Also, asset price fluctuations tend to be proportional to those prices. For example, a 50 dollar stock might experience daily price fluctuations on the order of one dollar while a 200 dollar stock might experience daily price fluctuations on the order of four dollars. This is not reflected by arithmetic random walks, whose standard deviations don't increase with the value of the process. For these reasons, geometric random walks often provide superior modeling of asset prices over time.

A geometric random walk is technically not a random walk, at least according to the general definition given above. It is a strictly positive stochastic process whose log returns follow a Gaussian white noise. This can be expressed as

 log( tR / t–1R ) = + tN [4]

where, again, the tN are independent and identically distributed standard normal random variables, and is constant.

All the above concepts generalize naturally to multivariate random walks.