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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. Model in probability theory. For the martingale betting strategy, see martingale betting system. Main article: Stopping time. Azuma's inequality Brownian motion Doob martingale Doob's martingale convergence theorems Doob's martingale inequality Local martingale Markov chain Markov property Martingale betting system Martingale central limit theorem Martingale difference sequence Martingale representation theorem Semimartingale.
Money Management Strategies for Futures Traders. Wiley Finance. Electronic Journal for History of Probability and Statistics. Archived PDF from the original on Retrieved Oxford University Press. Stochastic processes. Bernoulli process Branching process Chinese restaurant process Galton—Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy.
It only takes a minute to sign up. I was reading the Martingale betting system on wikipedia and there is something not completely clear to me. The article says at some point. However, the gambler's expected value does indeed remain zero because the small probability that he will suffer a catastrophic loss exactly balances with his expected gain.
It is widely believed that casinos instituted betting limits specifically to stop Martingale players, but in reality the assumptions behind the strategy are unsound. Players using the Martingale system do not have any long-term mathematical advantage over any other betting system or even randomly placed bets. I don't quite understand this. Thus, to me it seems that it should be rare , assuming that the probability of head is.
The probability that you get e. It'll happen to you eventually. I wrote a quick script in Matlab to simulate players following the Martingale strategy. I started players with a fortune of each, and had them all play the martingale redoubling strategy. Here are the results of one run:. As you can see, there are only 9 players who make it past 2, time steps, only 5 who make it past 4, and only 3 who make it past 6, By timestep 10, everyone has gone broke.
It's a common psychological bias to discount low-probability events when making decisions even when you know about the effect! Imagine a huge number of martingale players all starting to bet at the same time. The gambler's expected value does indeed remain zero. Whether the gambler makes 1 bet, or bets doesn't affect this. However what the Martingale betting system does change is the probability distribution of how much you win or lose. If you bet the same amount every time you are more likely to win or lose a little bit.
This is the probability distribution if you instead use the martingale betting system, and make 64 strings of bets. Sign up to join this community. The best answers are voted up and rise to the top. On Martingale betting system Ask Question. Asked 9 years, 2 months ago. Active 4 years, 10 months ago.