Studying random walks using martingales and stopping times.
Random Walk Theory in Finance. Perhaps the best and most widely known application of random walk theory is in finance. Random walk theory was first popularized by the 1973 book A Random Walk Down Wall Street by Burton Malkiel, an economics professor at Princeton University. The crux of the theory is that the price fluctuations of any given stock constitute a random walk, and therefore, future.
Random walk consists of a bunch of a sequence of IID random variables. The random walk consists of the sequence of partial sums of those random variables. And the question is if this random walk is taking place and you have two thresholds, one at alpha and one of beta-- and beta is below 0 and alpha is above 0-- and you start at 0, of course, and the question is when do you cross one of these.
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence, given all prior values, is equal to the present value.
In this subsection, we consider a number of applications of the martingale convergence theorems. One indication of the importance of martingale theory is the fact that many of the classical theorems of probability have simple and elegant proofs when formulated in terms of martingales. Simple Random Walk.
So RWH is a hypothesis which is consistent with EMH. If every piece of information is being priced in continuously, and you cannot predict what information will become available, then from your standpoint the price follows a random walk. On martingales: The stock itself is never a martingale in an efficient market. That is a popular misconception.
A martingale is a random walk, but not every random walk is a martingale. A Brownian random walk is a martingale if it does not have drift. Also, a martingale does not have to be a Markov process. EMH is not directly related to martingales.