Why Martingales are so important?

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I'm studing probability, an I can see martingale everywhere. I can work with them, but I don't really see in what they are so important everywhere.



I know that $M_n$ is a martingale wrt the filtration $mathcal F_n$ if $M_n$ is $mathcal F_n-$measurable and if $$mathbb E[M_n+1mid mathcal F_n]=M_n.$$
In what this definition makes them so important ? And why are they everywhere ?




probability-theory martingales




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  • 5




    Do you know how martingales originated? They came from gambling. They are models for what you call 'fair games'. However, they eventually became a powerful tool for proving theorems in Probability Theory. One of the nicest almost sure convergence theorems in Probability Theory is the Martingale Convergence Theorem.
    – Kavi Rama Murthy
    Aug 2 at 8:40










  • See here: math.stackexchange.com/questions/908908/…
    – Math1000
    Aug 2 at 13:06














up vote
2
down vote

favorite
1












I'm studing probability, an I can see martingale everywhere. I can work with them, but I don't really see in what they are so important everywhere.



I know that $M_n$ is a martingale wrt the filtration $mathcal F_n$ if $M_n$ is $mathcal F_n-$measurable and if $$mathbb E[M_n+1mid mathcal F_n]=M_n.$$
In what this definition makes them so important ? And why are they everywhere ?




probability-theory martingales




share|cite|improve this question

















  • 5




    Do you know how martingales originated? They came from gambling. They are models for what you call 'fair games'. However, they eventually became a powerful tool for proving theorems in Probability Theory. One of the nicest almost sure convergence theorems in Probability Theory is the Martingale Convergence Theorem.
    – Kavi Rama Murthy
    Aug 2 at 8:40










  • See here: math.stackexchange.com/questions/908908/…
    – Math1000
    Aug 2 at 13:06












up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





I'm studing probability, an I can see martingale everywhere. I can work with them, but I don't really see in what they are so important everywhere.



I know that $M_n$ is a martingale wrt the filtration $mathcal F_n$ if $M_n$ is $mathcal F_n-$measurable and if $$mathbb E[M_n+1mid mathcal F_n]=M_n.$$
In what this definition makes them so important ? And why are they everywhere ?




probability-theory martingales




share|cite|improve this question













I'm studing probability, an I can see martingale everywhere. I can work with them, but I don't really see in what they are so important everywhere.



I know that $M_n$ is a martingale wrt the filtration $mathcal F_n$ if $M_n$ is $mathcal F_n-$measurable and if $$mathbb E[M_n+1mid mathcal F_n]=M_n.$$
In what this definition makes them so important ? And why are they everywhere ?





probability-theory martingales





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 2 at 13:05







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  • 5




    Do you know how martingales originated? They came from gambling. They are models for what you call 'fair games'. However, they eventually became a powerful tool for proving theorems in Probability Theory. One of the nicest almost sure convergence theorems in Probability Theory is the Martingale Convergence Theorem.
    – Kavi Rama Murthy
    Aug 2 at 8:40










  • See here: math.stackexchange.com/questions/908908/…
    – Math1000
    Aug 2 at 13:06












  • 5




    Do you know how martingales originated? They came from gambling. They are models for what you call 'fair games'. However, they eventually became a powerful tool for proving theorems in Probability Theory. One of the nicest almost sure convergence theorems in Probability Theory is the Martingale Convergence Theorem.
    – Kavi Rama Murthy
    Aug 2 at 8:40










  • See here: math.stackexchange.com/questions/908908/…
    – Math1000
    Aug 2 at 13:06







5




5




Do you know how martingales originated? They came from gambling. They are models for what you call 'fair games'. However, they eventually became a powerful tool for proving theorems in Probability Theory. One of the nicest almost sure convergence theorems in Probability Theory is the Martingale Convergence Theorem.
– Kavi Rama Murthy
Aug 2 at 8:40




Do you know how martingales originated? They came from gambling. They are models for what you call 'fair games'. However, they eventually became a powerful tool for proving theorems in Probability Theory. One of the nicest almost sure convergence theorems in Probability Theory is the Martingale Convergence Theorem.
– Kavi Rama Murthy
Aug 2 at 8:40












See here: math.stackexchange.com/questions/908908/…
– Math1000
Aug 2 at 13:06




See here: math.stackexchange.com/questions/908908/…
– Math1000
Aug 2 at 13:06















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