I currently have an understanding of what a filtration is and will illustrate this through an example. From this example, I will convey what my idea of natural filtration is. My question is similar to this one which has not yet been answered. Before reading one, what I seek to know is:
- Is my interpretation of filtration correct? I am looking to build intuition. (Example 1)
- How does a filtration differ from a natural filtration? (Example 2)
- Does filtration differ from process history?
Example 1: Let's consider the generic binomial model of stock prices moving over $N$ periods where it can go up $U$ or down $D$ with probability $p$ and $1-p$, respectively. If the process is stock price $S_n$ then denote the stochastic process as $\{S_n:n=0,1,2,\dots,N\}$. For convenience, define the following Cartesian product
$$
\mathcal{C}_k := \underbrace{\{U,D\} \times \{U,D\} \times \dots \times \{U,D\}}_{k\mbox{ times}}
$$
from which the set of outcomes is $\Omega = \mathcal{C}_N $. The first filtration is trivial in that an elementary event occurs or does not such that
$$
\mathcal{F}_{0} = \{ \varnothing,\Omega \}.
$$
In the next trading period/iteration, the first stock movement will be revealed. The process will be $\mathcal{F}_1$-adapted in that the future is unknown and cannot be revealed as this is not information that the process has. At this point $\mathcal{F}_1$ is the information of the process and is given as
$$
\mathcal{F}_1 = \{U\}\times \mathcal{C}_{N-1}\cup\{D\}\times \mathcal{C}_{N-1} \cup\mathcal{F}_0
$$
which satisfies $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \dots \subseteq \mathcal{F}_{N-1} \subseteq \mathcal{F}_N$. Inductive reasoning deduces the following:
$$
\mathcal{F}_n = \mathcal{F}_{n-1} \cup \bigcup_{\sigma \in \mathcal{C}_n} \sigma \times \mathcal{C}_{N-n}, \quad n = 1,2,\dots,N .
$$
From the above, can we interpret $\mathcal{F}_n$ as follows. Lets ignore the inclusion of $\mathcal{F}_{n-1}$ first and focus on the part that is new. This new part is the uncertain behaviour we can expect over $n+1,n+2,\dots,N$ (no information part) for every possible realised path/history up to $n$ (the information part). Including $\mathcal{F}_{n-1}$ is a bit unclear to me, but I think it is there to denote that the $n^{th}$ trading period did not occur. Am I right in saying so?
Example 2: Now we get to the natural filtration. Some texts describe is as the minimal filtration. This gives me the idea that the filtration from example 1 is then a natural filtration. Planet Math says that a stochastic process generates this minimal filtration. This is maybe why it is difficult to imagine examples that aren't natural filtrations because we are thinking of how the stochastic process evolves and building an example of the filtration from that. Such reasoning will inevitably give a natural filtration. Perhaps an example of a filtration that is not a natural filtration can be built using a Hidden Markov Model with underlying states $x_n$ and observations $y_n$? The natural filtration will pertain to $x_n$ but the non-natural filtration might describe $(x_n,y_n)$.
Lastly, it seems to me that a filtration is different from process histories. We can consider each $\sigma \in \mathcal{C}_n$ to be a process history? Hence, a filtration considers all uncertain futures for each possible realised process history.
Edit: The following perhaps appears to be a bit funky:
$$
\mathcal{F}_1 = \{U\}\times \mathcal{C}_{N-1}\cup\{D\}\times \mathcal{C}_{N-1} \cup\mathcal{F}_0
$$
$$
\mathcal{F}_n = \mathcal{F}_{n-1} \cup \bigcup_{\sigma \in \mathcal{C}_n} \sigma \times \mathcal{C}_{N-n}, \quad n = 1,2,\dots,N .
$$
The convention is unconventional with respect to $\mathcal{C}_n$. The reason behind this is that it is how I went about reasoning on how one might write a general program to output filtrations for discrete state stochastic processes evolving over discrete points. To see how it works, consider that a trajectory has been observed at $n=3$ such that $\sigma_3 = \{U,U,D\}$. The stochastic process is of length $N=4$ such that $\mathcal{C}_{4-3} = \{ U,D\}$. Hence, $\mathcal{F}_4 = \mathcal{F}_3 \cup \sigma_{3}\times \{U\} \cup \sigma_{3}\times\{D\} = \mathcal{F}_3 \cup \{U,U,D,U\} \cup \{U,U,D,D\}$. Furthermore, it seems that I have mistaken an observed trajectory $\sigma_n$ to be the process history which is actually $\mathcal{F}_n$.
Update if anyone with an "applied" background as opposed to one using rigorous mathematics, please look at this gem of a paper. I found it after asking this question and I really feel that it has expanded my understanding.
Best Answer
$$ \mathcal{F}_1 = \{U\}\times \mathcal{C}_{N-1}\cup\{D\}\times \mathcal{C}_{N-1} \cup\mathcal{F}_0 $$ quite odd, in particular because of the $\times \mathcal{C}_{N-1}$. Better is in my opinion $$ \mathcal{F}_1 = \{\varnothing,U,D,\Omega\} $$ where $U$ and $D$ represent the events that $S_1$ has gone up, resp. down.
A typical example of a filtration that is larger than the natural one is the filtration of a multi-dimensional Brownian motion. Every one-dimensional component of such a BM is a BM with respect to its natural filtration and with respect to the larger filtration.
A standard treatment of a filtration for a discrete binomial process can be obtained as follows:
Fix a maturity $N$ and let $\Omega$ be the set that consists of all "paths" $\{U,D\}^N$ i.e. all $N$-tuples with values in $\{U,D\}$. The final $\sigma$-algebra ${\cal F}={\cal F}_N$ on this finite set of paths is clearly the set of all subsets of $\Omega$ i.e. the power set ${\cal P}(\Omega)$.
Since the above applies to all maturities $n$ strictly before $N$ we have a $\sigma$-algebra ${\cal F}_n={\cal P}(\{U,D\}^n)$ for each $n$. It is somewhat of a formal pseudo problem that a priory the elements of ${\cal F}_n$ are not subsets of $\Omega=\{U,D\}^N$.
A way around this is to consider an event $A\in{\cal F}_N$ and take the maximal index $n$ that is needed to uniquely describe it. For example $$ A=\{S_1=U,S_2-S_1=D\}. $$ This event belongs to ${\cal F}_2$.
Why do you want to write a program to generate the discrete filtration $({\cal F}_n)_{n=1,...,N}$ ? If this is that important then the simple approach in 2. is the route that I would take.