The martingale hypothesis defines that the level of any variable in  is equal to the price of the same variable in t using all the past information set.
Analytically, the martingale is a stochastic process of if the conditions , and hold. If It represents the information set available at time t built on the past history of the variable, we obtain , or equivalently:  (using the iterated expectation law).
Note that the stochastic process  is also defined as fair game because the expected value of the variable in a defined interval, given the information set available, equals zero. Using an alternative definition, the expected variation, influenced by the past history of the variable, is zero. It follows that the probability of positive/negative variation of the variable is the same. 
Using the asset price as an example, we can say that the best forecast of tomorrow’s price is today’s price, where the "best" is understood as "with lower average square root".
The martingale hypothesis is generally associated with the efficient market theory. We can use the martingale to define the weak form of market efficiency: if the market is efficient, in a weak form, it is not possible to systematically generate return trading on the information of past prices. Hence, the expectation on a future variation of price influenced by the price history set must be equal to zero. Consequently, the more efficient the market, the more random the variation of price. The most efficient market is the one in which changes in prices are completely random and unpredictable.
The martingale hypothesis is not related in any way to the risk hypothesis. The trade-off between expected return and risk is a fundamental pillar of the modern finance theory. Even with this limit, the martingale hypothesis is widely used in the pricing theory after checking for the risk correction model.
Campbell, J.Y., Lo, W.A., MacKinlay, A.C., 1997, The Econometrics of Financial Markets, Princeton University Press, Princeton New Jersey;
Becchetti, L., Ciciretti, R., Trenta, U., 2007, Modelli di Asset Pricing I: Titoli Azionari, in Il Sistema Finanziario Internazionale, Michele Bagella, a cura di, Giappichelli, Torino.
Editor: Rocco CICIRETTI