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Much of the original development of the theory was done by Joseph Leo Doob among others. Part of the motivation for that work was to show the impossibility of successful betting strategies in games of chance.

A basic definition of a discrete-time martingale is a discrete-time stochastic process i. That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.

Similarly, a continuous-time martingale with respect to the stochastic process X t is a stochastic process Y t such that for all t. It is important to note that the property of being a martingale involves both the filtration and the probability measure with respect to which the expectations are taken.

These definitions reflect a relationship between martingale theory and potential theory , which is the study of harmonic functions.

Given a Brownian motion process W t and a harmonic function f , the resulting process f W t is also a martingale.

The intuition behind the definition is that at any particular time t , you can look at the sequence so far and tell if it is time to stop. An example in real life might be the time at which a gambler leaves the gambling table, which might be a function of their previous winnings for example, he might leave only when he goes broke , but he can't choose to go or stay based on the outcome of games that haven't been played yet.

That is a weaker condition than the one appearing in the paragraph above, but is strong enough to serve in some of the proofs in which stopping times are used.

The concept of a stopped martingale leads to a series of important theorems, including, for example, the optional stopping theorem which states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.

From Wikipedia, the free encyclopedia. To understand the basics behind the martingale strategy, let's look at an example.

There is an equal probability that the coin will land on heads or tails. Each flip is an independent random variable , which means that the previous flip does not impact the next flip.

The strategy is based on the premise that only one trade is needed to turn your account around. Unfortunately, it lands on tails again.

As you can see, all you needed was one winner to get back all of your previous losses. However, let's consider what happens when you hit a losing streak:.

You do not have enough money to double down, and the best you can do is bet it all. You then go down to zero when you lose, so no combination of strategy and good luck can save you.

You may think that the long string of losses, such as in the above example, would represent unusually bad luck. But when you trade currencies , they tend to trend, and trends can last a long time.

The trend is your friend until it ends. The key with a martingale strategy, when applied to the trade, is that by "doubling down" you lower your average entry price.

As the price moves lower and you add four lots, you only need it to rally to 1. The more lots you add, the lower your average entry price.

On the other hand, you only need the currency pair to rally to 1. This example also provides a clear example of why significant amounts of capital are needed.

The currency should eventually turn, but you may not have enough money to stay in the market long enough to achieve a successful end.

That is the downside to the martingale strategy. One of the reasons the martingale strategy is so popular in the currency market is that currencies, unlike stocks , rarely drop to zero.

Although companies can easily go bankrupt, most countries only do so by choice. There will be times when a currency falls in value.

However, even in cases of a sharp decline , the currency's value rarely reaches zero. The FX market also offers another advantage that makes it more attractive for traders who have the capital to follow the martingale strategy.

The ability to earn interest allows traders to offset a portion of their losses with interest income. That means an astute martingale trader may want to use the strategy on currency pairs in the direction of positive carry.

The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance.

In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables , an assumption which is valid in many realistic situations.

It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet.

In most casino games, the expected value of any individual bet is negative, so the sum of many negative numbers will also always be negative. The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets which is also true in practice.

The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the optional stopping theorem.

Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler "resets" and is considered to have started a new round.

A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds.

Following is an analysis of the expected value of one round. Let q be the probability of losing e. Let B be the amount of the initial bet. Let n be the finite number of bets the gambler can afford to lose.

The probability that the gambler will lose all n bets is q n. When all bets lose, the total loss is. In all other cases, the gambler wins the initial bet B.

Thus, the expected profit per round is. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round.

Increasing the size of wager for each round per the martingale system only serves to increase the average loss. Suppose a gambler has a 63 unit gambling bankroll.

The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, taking k as the number of preceding consecutive losses, the player will always bet 2 k units.

With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point. Once this win is achieved, the gambler restarts the system with a 1 unit bet.