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I know the natural filtration intuitively represents the history of the process as the process evolves over time, and hence can be used to talk about conditional probabilities and conditional expectations (say).

But what's the motivation/significance of the concept of a filtration (i.e. an increasing sequence of sub-sigma algebras) in general?

Put differently, why would one be interested in filtrations that are not natural filtrations?

Thank you in advance!

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The natural filtration, as you said, refers to the filtration of a particular process. The filtration generated by two different processes is not necessarily the same as the natural filtration of a process. For example let's take a one step model $$ \Omega=\{\omega_1,\omega_2,\omega_3\}\\ \mathcal A=\mathcal P(\Omega)\\ X_0=Y_0=0\\ X_1(\omega_1)=0\\ X_1(\omega_2)=0\\ X_1(\omega_3)=1\\ Y_1(\omega_1)=0\\ Y_1(\omega_2)=1\\ Y_1(\omega_3)=1\\ \mathcal F^X_1=\mathcal \{\{\omega_1,\omega_2\},\{\omega_3\},\Omega, \emptyset \}\\ \mathcal F^Y_1=\mathcal \{\{\omega_2,\omega_3\},\{\omega_1\},\Omega, \emptyset \}\\ \mathcal F_1=\mathcal P(\Omega)\\ \mathcal F_0^X=\mathcal F_0^Y=\mathcal F_0=\{\Omega,\emptyset\} $$

The notion of filtration is related to the notion of measurability, i.e. to know the law of probability of a random variable, or process. In the example above, if the sigma algebra was $\mathcal A= \mathcal F_1^Y$, you could define the process $X$ but you could not measure it, (you could not know it's law).

You could also define the process $$ Z_0(\omega_1)=1\\ Z_0(\Omega \backslash \{\omega_1\})=0\\ Z_1=1 $$ $Z_0$ is $\mathcal A$-measurable. Therefore it makes sense as a random variable. However, $Z$ is not adapted to the filtration $\mathcal F=\sigma(X,Y)$

$Z$ is like a way to know the future at 0. In financial markets one filtration represents the flow of information that becomes public at a given point in time. It is always increasing. Imagine that there was a variable that gave the result of a dice before throwing the dice. Impossible right? It would not be adapted to the natural filtration of the result of the dice. In this sense, a filtration is defined by all publicly available information. Therefore, the filtration is used to separate different classes of processes. Those that are adapted to the filtration and those that aren't. Moreover, when you define a martingale, you must define it with regards to a certain filtration. Stopping times also must be defined with regards to a certain filtration. Usually when you say $X$ is a martingale and don't specify a filtration it means with regards to it's natural filtration, because if a process is a martingale for any filtration, it is a martingale for it's natural filtration though.

This article is really good on the subject Check out also the Wikipedia page on filtrations

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  • $\begingroup$ Thank you for taking out the time to write such an amazing answer. Although it explains the concept of natural filtration beautifully, I am afraid it does not directly answer the question I asked. Interpreting a filtration as information flow is explicitly justified only when it's a natural filtration. But in general, I am not sure we can do that. If we can't, then using a filtration to 'classify processes' would not work, as classification into adapted and previsible processes depends upon the interpretation of a filtration as information flow with time. Please correct me if I am wrong. $\endgroup$ – Dhruv Gupta Mar 19 at 4:22
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    $\begingroup$ There is also an information flow for each individual, imagine you are on a desert island far away, you would get the news delayed. Your filtration is not a filtration natural to a process. It's just an information flow. $\endgroup$ – Thomas G. Mar 19 at 6:30
  • $\begingroup$ I also loved your comment in which you pointed out the world could have multiple processes (say X and Y) evolving over time, and so the information flow wouldn't be the natural filtration of any one of the process. You seemed to have edited that away. If you choose to put it up again, I would be sure to upvote it! $\endgroup$ – Dhruv Gupta Mar 20 at 4:42
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    $\begingroup$ If the world has only X and Y the information flow is a filtration that is not natural to X or Y. Bear un mind though that it is the natural filtration of the vector process (X,Y) $\endgroup$ – Thomas G. Mar 20 at 6:12
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Imagine a world where the filtration all available information is $(F_n)_{n\in \mathbb N}$ Imagine an individual that gets the information one period late. His information filtration is $(F'_n)_{n\in\mathbb N}$ such that $F'_n=F_{n-1}$. There is a priori no process for which $(F'_n)_{n\in\mathbb N}$ is natural.

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