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Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. A stochastic process is predictable with refer to the filtration $(\mathcal F_n)_n$ if $X_{n+1}\in \mathcal F_n$.

Could someone tel me what it mean exactly ? (except the fact that $X_{n+1}$ is $\mathcal F_n-$measurable). What is the intuition behind ? Would it be a sort of "simple function" for stochastic process ?

Bernard
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user380364
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  • It means based on the information at time $n$ you know what value the process will take at time $n+1$. It is literally predictable as its name suggests. A typical example would be a betting/gambling/trading strategy. – Calculon May 02 '18 at 08:22
  • @Calculon: would you have an example of such process ? – user380364 May 02 '18 at 08:24
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    A trading strategy would be an example. Today you decide what to buy and sell and you update your portfolio daily. So today you already know what your portfolio tomorrow will be like. Hence it is a predictable process. – Calculon May 02 '18 at 08:27
  • See here for intuition: https://math.stackexchange.com/questions/2190739/what-exactly-is-a-predictable-process

    and here for why predictable processes are useful: https://math.stackexchange.com/questions/1497704/why-predictable-processes

     – Math1000 May 02 '18 at 08:38

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