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1) Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $(\mathcal F_n)$ for $n\in \mathbb N$ a filtration.

A) First of all, I'm not clear with filtration. Why are they important ? Indeed, we say that a stochastic process $(X_n)$ is adapted if $X_n\in \mathcal F_n$ for all $n$. But why don't we take $\mathcal F_n=\mathcal F$ for all $n$, and then $X_n\in \mathcal F_n=\mathcal F$ for all $n$... this notion of filtration is not very relevant to me.

B) We say that $(Y_n)$ is predictable if $Y_n\in \mathcal F_{n+1}$ for all $n$. What does it really mean ? For example, if $(X_n)$ is the gain at the $n-$th game, then $(X_n)$ is adapted, whereas if $(B_n)$ is the bet at the $n-$th game, then it's predictable. I don't really know what it really mean.

2) Let $(\mathcal F_t)_t$ a filtration but with $t\in \mathbb R^+$.

A) We say that $(\mathcal F_t)_t$ is right continuous at $t$ if $\bigcap_{s>t}\mathcal F_s=\mathcal F_t$. What is the interpretation of this notion ? What it mean concretely ?

B) What is a predictable process for such a process ? Is $(X_t)$ predictable if $X_t\in \mathcal F_{t+\varepsilon }$ for all $1>\varepsilon >0$ ? Or maybe $X_t\in \bigcap_{s>t}\mathcal F_s$ ? But if the filtration is continuous, then predictable process and adapted process are the same... so I'm not sure it's a good definition.

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    $\begingroup$ A predictable processus is not $Y_n\in \mathcal F_{n+1}$ but $Y_{n+1}\in \mathcal F_n$, i.e. you can describe $Y_{n+1}$ with the information you have at $n$, i.e. you can predict it. In other word, you know what $Y$ is going to be tomorrow with the information you have today. Notice that $Y_{n}\in \mathcal F_{n+1}$ for all $n$, would mean : to describe $Y$ today, you need the information of tomorrow... not very efficient :-) $\endgroup$ – Surb Dec 7 '18 at 11:11
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You can think of $\mathcal F _n$ as the amount of information available at time $n$. In that sense, using the same $\mathcal F$ for each time would be a very dull process: at each time you know the full history (past and future) of the process.

To model, that you only have knowledge of the past, we use an increasing (i.e. more information) filtration.

To relate this to a predictable process $Y_t$, this means that at time $t$, given the information of the past ($\mathcal F_t$), you already know what will happen at the next timestep ($Y_{t+1}$). This would for example be a strategy depending on the outcome of time $t$, think roulette where $Y_t$ could model the decision to bet on black or red.

For continuity, note that one can read $\bigcap$ as a limit (or more precisely $\liminf$) for sets. For the predictability in this case, your intuition is not far off: We want to know the value of $Y_t$ for small increments. Thus we will require $Y_t$ to be adapted to the sigma-algebra generated by the left-continuous random variables (see for example https://en.wikipedia.org/wiki/Predictable_process#Continuous-time_process).

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