Martingale theory, a branch under stochastic process is a mathematical model for a fair wager where prediction of future winnings is impossible even knowing the past events. This name is derived from “la grande martingale”, where the bet after a loss will be twice as the previous one. This is done keeping in mind that when we win the bet all the incurred loss till then will be covered by that win. As this method delivers a sure win strategy at some point of time it ensures profit.
Paul Levy in 1934 introduced martingale concept into probability. But the name martingale was coined by Ville in 1939 by extending the martingale concept introduced by Levy to continuous martingales. Jason Leo Doob continued with the development of martingale theory. [1]
Martingale are different from non-martingales in that, a non-martingale in which the expectation of present and next outcome are the same can have its uncertainties of upcoming outcomes reduced with knowledge of earlier outcomes. By using winning strategy, the expectation of next outcome may be greater than the one now by using the knowledge of previous and present outcomes. But Martingales are a model of fair games. [2]
Definition of Martingale: [1]
Martingales can be said to be the model of fair games as it doesn’t take in account the previous outcomes in winning strategies decision. Here fair can be defined to be the game in which the better comes out with a mean win of zero. This criteria is based on average calculation. The criteria used here may sound good, but not good enough. In addition to having a final expectation of zero, the expectation should be zero when the bet began. To explain this we will consider tossing of coins. Let us assume that we have three coins in which...
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... occur. To explain the equation above we consider that F_n is a natural sigma field, which automatically implies that F_T=σ(X_1,X_2…X_T). The sequence of F_T is a random length sequence. So the elements in the sequence should be should be carefully considered. Here a set of value over which T is constant must be selected. As an example, we take a scenario where T be a time at which an outdoor game is cancelled at a certain place. An event of interest for us would be why the game was cancelled (snow, rain etc.). Let Λ be the event which caused the game to be cancelled. This event has already occurred. To make sure that the event Λ happened, we can either look the look in news or in newspaper. If it is mentioned in any of those on day n, then T=n. If there is a mention of snow or rain in those articles, then that gives the event {T=n}∩Λ, which belongs to F_n. [1]
consists of all information available to any market participant. Researchable task in this study is whether stock asset returns are predictable, which has been a question of great attention emerged with financial econometrics since the earliest times. Mathematical models of asset pricing have an unusually rich history as compared to every other aspect of economic analysis. For tests of return predictability, information set is defined as the past history of stock prices, company characteristics, market characteristics