Understanding stochastic processes is important for many fields (e.g., statistical physics, mathematical finance, generative AI), and a filtration is a foundational concept. Here is the Wikipedia definition:

Filtrations are totally ordered collections of subsets that are used to model the information that is available at a given point

This can be pretty confusing (it certainly was for me), so I’d like to share a concrete example to help with the intuition.

Think of a group of kids in a preschool (sorry, these are US-centric examples), where you observe them from afar, not getting to know the individual kids, but just looking at some overall metrics and statistics on their activities and accomplishments.

After some time, the children will move to different elementary schools. Eventually, they move to different high schools, perhaps colleges, etc. Then they get different jobs, maybe some will get a dog, etc. The point is that over time, they begin to have memberships in different overlapping classes (classes such as which elementary school, which job, whether pet ownership, …).

So in the end, each child will have a unique trajectory through time: a sequence of memberships in these overlapping classes. If we randomly pick a kid and ask what trajectory that child’s life will take, then the set of kids in that preschool corresponds to the set of all possible outcomes \(\Omega\). At time \(t_0\), there is a \(\sigma\)-algebra (also called a field in probability theory) \(\mathcal F_0\) that is a collection that contains two subsets of \(\Omega\), namely \(\Omega\) itself (i.e., the set of all kids in the preschool) and the empty set \(\emptyset\) (recall that a \(\sigma\)-algebra must be closed under countable unions and complements).

Suppose that at time \(t_1\), the children sort into different elementary schools. Then, there is a \(\sigma\)-algebra \(\mathcal{F}_1\) that is a collection of all the subsets of kids that went to the same elementary schools plus all countable unions and complements. And therefore \(\mathcal{F}_0 \subseteq \mathcal{F}_1\).

Next, suppose that after all the kids have grown up and finished their education, at time \(t_2\) they sort into different jobs. Then, there is a \(\sigma\)-algebra \(\mathcal{F}_2\) that is a collection of all the \(\Omega\) subsets generated by the job types plus all the subsets in \(\mathcal{F}_1\) plus countable unions and complements. And therefore \(\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \mathcal{F}_2\).

Proceeding this way, by considering all times when the kids do or do not become members of various classes, we see that the observables (random variables) at each time (elementary school at \(t_1\), job type at \(t_2\), …) generate \(\sigma\)-fields, and the sequence of observables from \(t_0\) through \(t\) is a stochastic process that generates the \(\sigma\)-field \(\mathcal{F}_t\). So we can construct a filtration \(\mathbb{F}\) that is the family \(\{\mathcal{F}_t\}\) of increasing \(\sigma\)-algebras.

At any time \(t\), the \(\sigma\)-algebra \(\mathcal{F}_t\) will contain all the subsets of \(\Omega\) that are measurable at that time \(t\). If an event (some subset of \(\Omega\)) is not contained in \(\mathcal{F}_t\), then that event is not measurable, which means that as of time \(t\) neither that event nor the complement of that event will have occurred. For example, at \(t_1\) none of the children have sorted into job types because that will not happen until \(t_2\).

To make this concrete, assume that the preschool has 6 kids, numbered from 1 to 6. Suppose that at \(t_1\), #1, #2, and #3 went to elementary school A, and the others went to elementary school B. Later at \(t_2\), #1 and #2 became teachers and the others went into different careers. Then at \(t_2\) the event \(\{1,2\}\) is measurable because it is contained in \(\mathcal{F}_2\) (i.e., “job=teacher” corresponds to one of the \(\Omega\) subsets generated by the “jobs” random variable). However, at \(t_2\) the event \(\{1\}\) is NOT measurable. This is because through time \(t_2\), #1 and #2 have taken the same trajectory. And therefore if we observe someone who went to elementary school A and later took a job as a teacher, we would not be able to decide whether that person is #1 or #2.

Share or comment on Mastodon