## Lecture 4. Filtrations

A stochastic process $(X_t)_{t \ge 0}$ may also be seen as a random system evolving in time. This system carries some information. More precisely, if one observes the paths of a stochastic process up to a time $t\ge 0$, one is able to decide if an event

$A \in \sigma( X_s,s\le t)$

has occured (here and in the sequel $\sigma( X_s,s\le t)$ denotes the smallest $\sigma$-field that makes all the random variables $\left\{ (X_{t_1}, \cdots, X_{t_n} ), 0 \le t_1 \le \cdots \le t_n \le t \right\}$ measurable). This notion of information carried by a stochastic process is modeled by filtrations.

Definition. Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space. A filtration $(\mathcal{F}_t)_{t \ge 0}$ is a non-decreasing family of sub-$\sigma$-algebras of $\mathcal{F}$.

As a basic example, if $(X_t)_{t \ge 0}$ is a stochastic process defined on $(\Omega, \mathcal{F},\mathbb{P})$,then

$\mathcal{F}_t=\sigma( X_s,s\le t)$

is a filtration. This filtration is called the natural filtration of the process $X$ and will often be denoted by $(\mathcal{F}^X_t)_{t \ge 0}$.

Definition. A stochastic process $(X_t)_{t \ge 0}$ is said to be adapted to a filtration $(\mathcal{F}_t)_{t \ge 0}$ if for every $t \ge 0$, the random variable $X_t$ is measurable with respect to $\mathcal{F}_t$.

Of course, a stochastic process is always adapted with respect to its natural filtration. We may observe that if a stochastic process $(X_t)_{t \ge 0}$ is adapted to a filtration $(\mathcal{F}_t)_{t \ge 0}$ and that if $\mathcal{F}_0$ contains all the subsets of $\mathcal{F}$ that have a zero probability, then every process $(\tilde{X}_t)_{t \ge 0}$ that satisfies

$\mathbb{P}(\tilde{X}_t=X_t)=1, \quad t \ge 0,$

is still adapted to the filtration $(\mathcal{F}_t)_{t \ge 0}$.

We previously defined the notion of measurability for a stochastic process. In order to take into account the dynamic aspect associated to a filtration, the notion of progressive measurability is needed.

Definition. A stochastic process $(X_t)_{t \ge 0}$ that is adapted to a filtration $(\mathcal{F}_t)_{t \ge 0}$, is said to be progressively measurable with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$ if for every $t \ge 0$,

$\forall A \in \mathcal{B}(\mathbb{R}), \{(s,\omega) \in [0,t] \times \Omega, X_s (\omega) \in A \} \in \mathcal{B}([0,t])\otimes \mathcal{F}_t.$

By using the diagonal method, it is possible to construct adapted but not progressively measurable processes. However, the next proposition whose proof is let as an exercise to the reader shows that an adapted and continuous stochastic process is atomically progressively measurable.

Proposition. A continuous stochastic process $(X_t)_{t \ge 0}$, that is adapted with respect to a filtration $(\mathcal{F}_t)_{t \ge 0}$, is also progressively measurable with respect to it.

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