Natural filtration

In the theory of stochastic processes in mathematics and statistics, the generated filtration or natural filtration associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time. It is in a sense the simplest filtration available for studying the given process: all information concerning the process, and only that information, is available in the natural filtration.

Formal definition

Let

$(\Omega ,{\mathcal {A}},P)$ be a probability space.

$(I,\leq )$ be a totally ordered index set. In many examples, the index set $I$ is the natural numbers $\mathbb {N}$ (possibly including 0) or an interval $[0,T]$ or $[0,+\infty ).$

$(S,\Sigma )$ be a measurable space. Often, the state space $S$ is often the real line $\mathbb {R}$ or Euclidean space $\mathbb {R} ^{n}.$

$X:I\times \Omega \to S$ be a stochastic process.

Then the natural filtration of ${\mathcal {A}}$ with respect to $X$ is defined to be the filtration $\mathbb {F} ^{X}={\mathcal {F}}_{\bullet }^{X}=({\mathcal {F}}_{i}^{X})_{i\in I}$ given by

{\mathcal {F}}_{i}^{X}=\sigma \left\{\left.X_{j}^{-1}(B)\right|j\in I,j\leq i,B\in \Sigma \right\},

i.e., the smallest σ-algebra on $\Omega$ that contains all pre-images of $\Sigma$ -measurable subsets of $S$ for "times" $j$ up to $i$ .

Any stochastic process $X$ is an adapted process with respect to its natural filtration.

Examples

Two examples are given below, the Bernoulli process and the Wiener process. The simpler example, the Bernoullii process, is treated somewhat awkwardly and verbosely, belabored, but using a notation that allows more direct contact with the Wiener process.

Bernoulli process

The Bernoulli process is the process $X$ of coin-flips. The sample space is $\Omega =\{0,1\}^{\mathbb {N} }=2^{\mathbb {N} },$ the set of all infinitely-long sequences of binary strings. A single point $\omega \in \Omega$ then specifies a single, specific infinitely long sequence. The index set $I=\mathbb {N}$ is the natural numbers. The state space is the set of symbols $S=\{H,T\}$ indicating heads or tails. Fixing $\omega$ to a specific sequence, $X(i,\omega )$ then indicates the $i$ 'th outcome of the coin-flip, heads or tails. The conventional notation for this process is $X_{i}=X(i,-),$ indicating that all possibilities should be considered at time $i.$

The sigma algebra on the state space contains four elements: $\Sigma =\{\varnothing ,\{H\},\{T\},S\}.$ The set $X_{j}^{-1}(B)$ for some $B\in \Sigma$ is then a cylinder set, consisting of all strings having an element of $B$ at location $j:$

X_{j}^{-1}(B)=\{\omega \in \Omega :X(j,\omega )=b{\text{ and }}b\in B\}

The filtration is then the sigma algebra generated by these cylinder sets; it is exactly as above:

{\mathcal {F}}_{i}=\sigma \left\{X_{j}^{-1}(B):j\leq i,B\in \Sigma \right\},

The sub-sigma-algebra ${\mathcal {F}}_{i}$ can be understood as the sigma algebra for which the first $i$ symbols of the process have been fixed, and all the remaining symbols are left indeterminate.

This can also be looked at from a "sideways" direction. The set

C(i,\omega )=\{\alpha \in \Omega :X(j,\alpha )=X(j,\omega ){\text{ for }}j\leq i\}

is a cylinder set, for which all points $\alpha \in C(i,\omega )$ match exactly $X(-,\omega )$ for the first $i$ coin-flips. Clearly, one has that $C(k,\omega )\subset C(i,\omega )$ whenever $k>i.$ That is, as more and more of the initial sequence is fixed, the corresponding cylinder sets become finer.

Let $A\in {\mathcal {A}}$ be one of the sets in the sigma algebra ${\mathcal {A}}.$ Cylinder sets can be defined in a corresponding manner:

C(i,A)=\{\alpha \in \Omega :X(j,\alpha )=X(j,\omega ){\text{ for }}j\leq i{\text{ and }}\omega \in A\}

Again, one has that $C(k,A)\subset C(i,A)$ whenever $k>i.$

The filtration can be understood to be

{\mathcal {F}}_{i}=\{C(i,A)|A\in {\mathcal {A}}\}

consisting of all sets for which the first $i$ outcomes have been fixed. As time progresses, the filtrations become finer, so that ${\mathcal {F}}_{i}\subset {\mathcal {F}}_{k}$ for $i<k.$

Wiener process

The Wiener process $X$ can be taken to be set in the classical Wiener space $\Omega =C_{0}([0,T])$ consisting of all continuous functions on the interval $I=[0,T].$ The state space $S$ can be taken to be Euclidean space: $S=\mathbb {R} ^{n}$ and $\Sigma ={\mathcal {B}}(\mathbb {R} ^{n})$ the standard Borel algebra on $\mathbb {R} ^{n}.$ The Wiener process is then $X:I\times \Omega \to S.$

The interpretation is that fixing a single point $\omega \in \Omega$ fixes a single continuous path $X(\cdot ,\omega ):I\to S.$ Unlike the Bernoulli process, however, it is not possible to construct the filtration out of the components

X_{t}^{-1}(B)=\{\omega \in \Omega :X(t,\omega )=x{\text{ and }}x\in B\}

for some $B\in {\mathcal {B}}(\mathbb {R} ^{n}).$ The primary issue is that $I$ is uncountable, and so one cannot perform a naive union of such sets, while also preserving continuity. However, the approach of fixing the initial portion of the path does follow through. By analogy, define

C(t,\omega )=\{\alpha \in \Omega :X(s,\alpha )=X(s,\omega ){\text{ for }}s\leq t\}

This consists of all continuous functions, that is, elements of $\Omega$ for which the initial segment $s\leq t$ exactly matches a selected sample function $X(\cdot ,\omega ).$ As before, one has that $C(u,\omega )\subset C(t,\omega )$ whenever $t<u,$ that is, the set becomes strictly finer as time increases.

Presuming that one has defined a sigma algebra ${\mathcal {A}}$ on the (classical) Wiener space, then for a given $A\in {\mathcal {A}},$ the corresponding cylinder can be defined as