Talk:Natural filtration

Examples needed

I'm trying to imagine what the natural filtration for the Wiener process is, and all I get in my imagination is just the normal sigma algebra on the reals. In particular, I'm trying to understand the Tanaka equation and in what sense the natural filtration doesn't work, isn't appropriate/does the wrong thing. 67.198.37.16 (talk) 04:40, 4 May 2025 (UTC)

This same question was raised on Stack Exchange in Feb 2016 and has not yet received an answer: What are the elements of a filtration generated by a Wiener process? I can't even figure out what it is that I don't understand about this. I cannot find anything in any search engine that explains any of this in any level of detail. 67.198.37.16 (talk) 16:26, 4 May 2025 (UTC)

Sigh. I'm stupid. The article Stochastic process provides the needed silly insight. It says this:

A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if $\{X(t,\omega ):t\in T\}$ is a stochastic process, then for any point $\omega \in \Omega$, the mapping

$X(\cdot ,\omega ):T\rightarrow S$

is called a sample function, a realization, or, particularly when $T$ is interpreted as time, a sample path of the stochastic process $\{X(t,\omega ):t\in T\}$. This means that for a fixed $\omega \in \Omega$, there exists a sample function that maps the index set $T$ to the state space $S$. Other names for a sample function of a stochastic process include trajectory, path function.

This tells us how to construct the filtration. The points in $\Omega$ correspond to paths. The Borel sets in the sigma algebra in $\Omega$ correspond to collections of paths. Some of those Borel sets will encode a set of paths that are identical up to some time $s<T$ and then diverge every-which way from there. Obviously, if $s<u<T$ then the Borel set containing all paths that are identical up to time $u$ is going to be contained inside the Borel set for paths up to time $s$. Thus, the collection of Borel sets up to time $u$ is necessarily a refinement of those up to time $s$ and those at $s$ are coarser. Duhh. So this is pretty easy, once you see it. What I don't have is some good elegant notation to say the above with formulas. It's some cylinder set, but at this time, I have not seen any text (or Wikipedia page) that gives notation for "the cylinder set for which all paths are identical up to time s, and then can be anything after that." 67.198.37.16 (talk) 06:44, 9 May 2025 (UTC)