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I am currently studying results from probability theory in a continuous time setting and I keep reading the term standard filtration (right continous and complete). I understand the definition, but I don't really get the intuition behind it. Can anyone help my understanding by explaining why we considere the standard filtration and where filtrations without right continuity and completeness don't do the job any more?

Many thanks in advance!

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Let $X_t$ be some continuous-time stochastic process (say, Brownian motion). Then the standard filtration $\mathcal{F}_s$ is the sigma algebra generated by the family $\{X_t\}$ for all $t\leq s$.

Intuitively, $\mathcal{F}_s$ is all the "information" available to us up to time $s$. In other words, it is the "history" of the process up to time $s$.

As a concrete example, let $X_t$ be the stock price of some company at time $t$. This is a continuous time stochastic process. Then $\mathcal{F}_s$ is just the history of the price up to time $s$. If something is measurable with respect to $\mathcal{F}_s$, it only requires knowledge up to time $s$, and not the future.

Hope this helps, let me know if I can clarify further.

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