Filtration

Definition

Consider a probability space \fs4 (\Omega, \mathscr{F}, P). A filtration is a collection of non-decreasing \fs4 \sigma-algebras

\fs4 \mathbb{F} = \{\mathscr{F}_t : t \geq 0 \} = \{\mathscr{F}_t\}_{t \geq 0} \text{ with } \mathscr{F}_{t_1} \subseteq \mathscr{F}_{t_2} \text{ for all } t_1 \leq t_2

The quadruple \fs4 (\Omega, \mathbb{F}, \mathscr{F}, P) is called a filtered space. A process \fs4 X_t is \fs4 \mathscr{F}_t-adapted if all \fs4 X_t are \fs4 \mathscr{F}_t-measurable.

The natural filtration of a process \fs4 X_t (or the generated filtration) is the one generated by the collection of random variables \fs4 \{X_s : 0\leq s \leq t\}. We denote this filtration by \fs4 \mathscr{F}_t = \sigma(X_s:0\leq s \leq t), and it is the smallest filtration that makes \fs4 X_t an adapted process.

Intuition of filtration

The natural filtration keeps the information we gather by observing the process \fs4 X_t up to time \fs4 t. The experiment has chosen a trajectory \fs4 \omega that has determined the complete path \fs4 \{X_t : t\ge 0\} for us, but this path has not been completely revealed to us.

Our information consists only of the part \fs4 \{X_s : 0\leq s \le t \}. Based on this information we cannot pinpoint precisely which \fs4 \omega the experiment has selected, but we can tell with certainty if \fs4 \omega belongs on some specific subsets of \fs4 \Omega. These sets make up the natural filtration \fs4 \mathscr{F}_t, and of course \fs4 X_t is adapted

A process \fs4 Y_t(\omega) will be \fs4 \mathscr{F}_t-adapted if we can ascertain with certainty the value \fs4 Y_t by observing \fs4 X_t. In particular, a \fs4 Y_t will be adapted if \fs4 Y_t = f(\{X_s : 0\leq s\leq t\}) for all \fs4 t

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