Martingale Test

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Martingale Test

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Martingale Test

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Have you forgotten your login? Journal articles. Florian Bollotte AuthorId : Author. Hide details. Abstract : This paper examines how statistical tests for martingale hypothesis can be applied to au- dit a risk-neutral Economic Scenarios Generator ESG.

The martingale test usually used to appreciate the risk-neutrality of the generated scenarios consists in testing that the mean of discounted asset prices is constant over time, which is a necessary condition for a martingale process.

The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake.

As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing.

However, the exponential growth of the bets eventually bankrupts its users due to finite bankrolls. Stopped Brownian motion , which is a martingale process, can be used to model the trajectory of such games.

The term "martingale" was introduced later by Ville , who also extended the definition to continuous martingales.

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. For the martingale betting strategy, see martingale betting system.

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The martingale test usually used to appreciate the risk-neutrality of the generated scenarios consists in testing that the mean of discounted asset prices is constant over time, which is a necessary condition for a martingale process.

Although we have not found many studies that refer to statistical martingale tests in the framework of ESG, we have noticed that this approach can be very useful from the point of view of the actuarial function, allowing for example to detect implementation errors in the ESG and provide results more easily interpretable in sensitivity analysis than the standard mar- tingale test.

Nevertheless, their application can raise many questions for the actuarial function, especially as concerns the choice of the tests retained, the ESG variable asset to which we apply the test and the acceptance threshold, and we will thus propose some recommendations for the use of these tests within a risk neutral ESG.

Document type : Journal articles. Identifiers HAL Id : hal, version 1. Metrics Record views. 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. 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 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 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 Biased Maximal entropy.

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Martingale Test

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