Increasing sequence of sigma algebras

[Stephen Tashi]

The explanation of a continuous Markov process X(t) defines an indexed collection of sigma algebras by \mathcal{F}_t = \sigma\{ X(s): s < t\} and this collection is said to be increasing with respect to the index t.

I'm trying to understand why the notation used for set inclusion is used to express the relation of "increasing" for a collection of sigma algebras.

A straightforward approach is to think of a set of sigma algebras that are each a collection of subsets of the same set and to define the concept of sub-sigma algebra in terms of one collection of sets being a subset of another collection of sets.

However, don't \mathcal{F}_t and \mathcal{F}_s denote sigma algebras defined on different sets when s \ne t ? I think of \mathcal{F}_t as being a sigma algebra of subsets of (only) the set of all trajectories of the process up to time t. \ If s > t then isn't \mathcal{F}_s a sigma algebra of subsets of a different set of trajectories?

 

[andrewkirk]

Every $\mathscr{F}_t$ is a subset of the powerset of $\Omega$, which is the set of all full trajectories, from the earliest time to the latest time.

The difference between $\mathscr{F}_s$ and $\mathscr{F}_t$ where $s<t$ is that the former groups together all full trajectories that have the same path up to time $s$ and the latter does the same in relation to the later time $t$. So the latter is a finer subdivision because it has the extra info that emerges between times $s$ and $t$.

 

[Stephen Tashi]

Every $\mathscr{F}_t$ is a subset of the powerset of $\Omega$, which is the set of all full trajectories, from the earliest time to the latest time.

Ok

The difference between $\mathscr{F}_s$ and $\mathscr{F}_t$ where $s<t$ is that the former groups together all full trajectories that have the same path up to time $s$ and the latter does the same in relation to the later time $t$.

What is the technical implementation of the concept of "groups together"?

So the latter is a finer subdivision because it has the extra info that emerges between times $s$ and $t$.

I understand the concept of "extra info", but I don't see what mathematical object is being subdivided.

 

[andrewkirk]

Let $t_0$ be the earliest time and $T$ the latest time, and $\Omega$ be the set of all possible paths traced out by the process between these two. $\Omega$ is called the sample space. $\mathscr{F}$ (note the lack of subscript) is a subset of the powerset of $\Omega$, that defines the set of all measurable subsets of $\Omega$. We will denote paths in $\Omega$ by $\omega$.

Denote the value of path $\omega$ at time $t$ by $X(t,\omega)$.

Then for every time $t$ define an equivalence relation $\sim_t$ on $\Omega$ by:

$$\omega_1\sim_t\omega_2\textrm{ iff }\forall s\in[t_0,t]\ X(s,\omega_1)=X(s,\omega_2)$$

That is, the paths $\omega_1,\omega_2$ are equivalent by relation $\sim_t$ iff they are identical up to time $t$ (They may diverge after that).

Then, if we define $\mathscr{C}_u$ as the partition of $\Omega$ arising from the equivalence relation $\sim_u$, we can define $\mathscr{F}_t\equiv\bigcup{u=t_0}^t\mathscr{C}_u$. [That last sentence has been corrected, based on observations by Stephen Tashi]

The next para, italicised and enclosed in square brackets [], is wrong, so ignore it. I've left it in just to avoid the change causing confusion about what was originally written
[So if $S\in\mathscr{F}_t$ then all paths in $S$ are identical up to time $t$. If $S_1,S_2\in\mathscr{F}_t$ and $S_1\neq S_2$ then every path in $S_1$ must differ from every path in $S_2$ for at least one time in the range $[t_0,t]$.]

It is easy to deduce from this that $\mathscr{F}_t$ is a sigma algebra that is a subset (subalgebra) of $\mathscr{F}$ and that $\mathscr{F}_s\subseteq\mathscr{F}_t$ if $s<t$.

The sequence of sigma algebras $\{\mathscr{F}_t\}_{t\in[t_0,T]}$ is called a filtration on the measure space $(\Omega,\mathscr{F})$.

 

[Stephen Tashi]

It is easy to deduce form this that $\mathscr{F}_t$ is a sigma algebra that is a subset (subalgebra) of $\mathscr{F}$ and that $\mathscr{F}_s\subseteq\mathscr{F}_t$ if $s<t$.

If a \in \mathscr{F}_s then a can be written as an uncountable union of those members in \mathscr{F}_t that have the same trajectory as a on the interval [t_0,s], but how do I see that a can be expressed using the operations of a sigma algebra?

 

[andrewkirk]

how do I see that a a can be expressed using the operations of a sigma algebra?

Are you asking whether $a\in\mathscr{F}_s$ can be expressed in terms of elements of $\mathscr{F}_t$ using the operations that are closed within a sigma algebra? If so then it's easy, because $a$ is a member of $\mathscr{F}_t$ as well, since $\mathscr{F}_s\subseteq\mathscr{F}_t$. So the expression needed is simply $a=a$.

 

[Stephen Tashi]

If so then it's easy, because $a$ is a member of $\mathscr{F}_t$ as well,

The way I look at it intuitively is that \mathscr{F}_t is generated by a collection of sets that are each "smaller than a" or exclude a. \ Am I not visualizing this correctly? We did say that \mathscr{F}_t is formed by taking equivalence classes of sets that make a finer partition of \mathscr{F} than the partition used to define \mathscr{F}_s.

For example, pretending for the moment that we have a discrete index, an element of the partition used to define \mathscr{F}_s could be as set like a = all sets of the form \{1,5,7, ?, ?, ?, ... \} where "?" can be an arbitrary value. An element of the partition used to define \mathscr{F}_t could be something like b = all sets of the form \{1,5,7,6,4,?,?,? ...\}. We have b \subset a.

 

[andrewkirk]

For example, pretending for the moment that we have a discrete index, an element of the partition used to define $\mathscr{F}_s$ could be as set like $a$ = all sets of the form {1,5,7,?,?,?,...} where "?" can be an arbitrary value. An element of the partition used to define $\mathscr{F}_t$ could be something like $b$ = all sets of the form {1,5,7,6,4,?,?,?...}. We have ##b \subset a## .

Actually, this makes me realize that I didn't express it quite correctly in post 4, where I said we 'define $\mathscr{F}_t$ as the partition of $\Omega$ arising from the equivalence relation $\sim_t$.' Most sigma-algebras are not partitions, because they contain overlapping sets, and in this case they certainly do.

What I should have written is that, if we use $\mathscr{C}_u$ to denote the collection of sets that is the partition of $\Omega$ defined by $\sim_u$, then $\mathscr{F}_t$ is the sigma algebra generated by $\bigcup_{u=t_0}^t \mathscr{C}_u$.

So every new value of $t$ adds a new bunch of sets to the growing sigma algebra. Because the $t$ index is continuous, we don't get a nice intuitive feel for this, that enables us to imagine the sigma algebra being 'constructed' as $t$ increases, like we do with discrete indices, because there is no 'next' or 'last' $t$. But it's all perfectly well-defined and rigorous.

I hope that makes more sense. Sorry for misleading you.

 

[Stephen Tashi]

What I should have written is that, if we use $\mathscr{C}_u$ to denote the collection of sets that is the partition of $\Omega$ defined by $\sim_u$, then $\mathscr{F}_t=\bigcup_{u=t_0}^t \mathscr{C}_u$.

Do we need to say \mathscr{F}_t = the sigma algebra generated by \bigcup_{u=t_0}^t \mathscr{C}_u ?

But anyway, I get the basic idea now. Thank you.

 

[andrewkirk]

@Stephen Tashi Yes we do: another good catch.