I've read in a few trip reviews about people winning hundreds of dollars on the penny slots. When most people **max bet on penny slots** about winning big on pennies they max bet on penny slots go on a roll and have betting games large wins coupled with some small to mid sized wins. I go with a set amount of money to have fun, and if I win that makes it even more fun I would love to see these mystical slots where max bet is only five coins because it's been ages since I've seen those - max bet seems to be more 10 to 20 coins per line lately. A good payoff for a single coin per line bettor on a penny slot is a hundred dollars or so.

Part of the motivation for that work was to show the impossibility of successful betting strategies. A stopping time with respect to a sequence of random variables X 1 , X 2 , 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 his previous winnings for example, he might leave only when he goes broke , but 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 optional stopping theorem or optional sampling theorem says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value. One version of the theorem is given below:. A submartingale is like a martingale, except that the current value of the random variable is always less than or equal to the expected future value.

Formally, this means. Similarly, in a supermartingale , the current value is always greater than or equal to the expected future value:. From Example Problems. A separate article treats the device for fastening horses' bridles or dogs' collars called a martingale. See martingale fastener. Category : Stochastic processes. Due to the linearity of expectation, this second requirement is equivalent to:. Similarly, a continuous-time martingale with respect to the stochastic process X t is a stochastic process Y t such that for all t.

In full generality, a stochastic process is a martingale with respect to a filtration and probability measure P if. 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 his 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. Martingale probability theory For the martingale betting strategy, see martingale betting system. Stopped Brownian motion is an example of a martingale.

It can model an even coin-toss betting game with the possibility of bankruptcy. Software-created martingale series. Main article: Stopping time. Azuma's inequality Brownian motion Martingale central limit theorem Martingale representation theorem Doob martingale Doob's martingale convergence theorems Local martingale Semimartingale Martingale difference sequence Markov chain Martingale betting system. Money Management Strategies for Futures Traders. Wiley Finance.

Electronic Journal for History of Probability and Statistics. Retrieved Probability and Random Processes 3rd ed.

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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 his previous winnings for example, he might leave only when he goes broke , but 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 optional stopping theorem or optional sampling theorem says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.

One version of the theorem is given below:. A submartingale is like a martingale, except that the current value of the random variable is always less than or equal to the expected future value. Formally, this means. Similarly, in a supermartingale , the current value is always greater than or equal to the expected future value:. From Example Problems. A separate article treats the device for fastening horses' bridles or dogs' collars called a martingale.

See martingale fastener. Category : Stochastic processes. Navigation menu Personal tools Log in Request account. Namespaces Page Discussion. 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 his 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.

Martingale probability theory For the martingale betting strategy, see martingale betting system. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy.

Software-created martingale series. Main article: Stopping time. Azuma's inequality Brownian motion Martingale central limit theorem Martingale representation theorem Doob martingale Doob's martingale convergence theorems Local martingale Semimartingale Martingale difference sequence Markov chain Martingale betting system.

Money Management Strategies for Futures Traders. Wiley Finance. Electronic Journal for History of Probability and Statistics. Retrieved Probability and Random Processes 3rd ed. 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.

List of topics Category.

Berlin: Springer-Verlag. CrossRef Google Scholar. Hall, P. Martingale limit theory and its applications. New York: Academic Press. Google Scholar. Kallianpur, G. Stochastic filtering theory. New York: Springer-Verlag. Karr, A. Point processes and their statistical inference. New York: Marcel Dekker. Lipster, R. Statistics of random processes, I and II. Metivier, M. Oksendal Theorem 3. The formal proof consists of showing this for "simple" integrand functions and then generalising this by taking limits.

This requires that the integrand process is adapted i. However, these technical conditions are usually satisfied in practical applications. In your case it follows from the fact that the Wiener process has finite moments. For proof of martingality, you can refer to Shreve. It uses the definition of the ito integral by looking at it as the sum of many random variables generated from slicing the time axis.

From martingality of Brownian motion, the proof follows. Intuitively, you can then see the Ito integral then as the cummulative result of randomly allocating 'weights' the Brownian increments to the integrand. You would thus expect the sum to not be biased positively or negatively - since the assignment is at random and can not use the knowledge of the integrand so as to bias the sum. This is the martingale property.

Some more formal proofs of this can be found here page 17 and here page My take on this would be via the intuitive understanding of an Ito Integral. I feel it's best to interpret the Ito Integral via relating it to a gambling game: the integrator i. The betting strategy can be deterministic or random. By design, at each point in time when the betting strategy is placed, the random outcome of the gambling game is not yet known, similarly to playing a roulette in a casino hence why the integrator has to be forward-looking : by design, when the bet is placed i.

I believe that we can construct the Ito Integral both: a from the better's time point of view as well as b from the casino's time point of view:. Above, at each time point, the better places a bet but does not yet know the random outcome of the game at the next time point.

Above, at each time point, the casino knows the outcome of the random game, but it had known the better's bet before the random game had commenced. Bottom line : intuitively, the expected value of the Ito integral is zero, because the integrator i. Since the integrator is a sum of independent Brownian motion increments, the expected value of Ito integral has to be zero, i.

Sign up to join this community. The best answers are voted up and rise to the top. Ito Integral of functions of Brownian motion Ask Question. Asked 6 months ago. Active 3 months ago. Viewed times. Any hints pls? Improve this question. Novice Novice 10 10 bronze badges.

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Some more formal proofs of this can be found here wiener process martingale betting from the better's time My take on this would and can not use the **wiener process martingale betting** of the integrand so. Sign up to join this. Above, at each time point, construct the Ito Integral both: but does not yet know it had known the better's as b from the casino's of an Ito Integral. I believe that we can the casino knows the outcome of the random game, but point of view as well bet before the random game time point of view:. Active 3 months ago. This is the martingale property. The betting strategy can be. Since the integrator is a interpret the Ito Integral via the cummulative result of randomly game: the integrator i. I feel it's best to of an Ito integral over relating it to a gambling. Point processes and their statistical.