# Predictable process, measurable process, filtration (continuous and discrete)

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.

• 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 :-) – Surb Dec 7 '18 at 11:11

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 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).