Filtration, Adapted Process, and Natural Filtration

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathbb{R}$ the set of real numbers.

Filtration. A filtration on $(\Omega,\mathcal{F},P)$ is a family $(\mathcal{F}_t)_{t\ge0}$ of sub-$\sigma$-algebras of $\mathcal{F}$ indexed by the nonnegative real numbers such that

A probability space equipped with a filtration is called a filtered probability space, written $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$.

Adapted process. A stochastic process $X=(X_t)_{t\ge0}$ on $(\Omega,\mathcal{F},P)$ is adapted to the filtration $(\mathcal{F}_t)_{t\ge0}$ if for every $t\ge0$ the random variable $X_t$ is $\mathcal{F}_t$-measurable, that is, $X_t^{-1}(B)\in\mathcal{F}_t$ for every Borel set $B$.

Natural filtration. For a stochastic process $X=(X_t)_{t\ge0}$, the natural filtration of $X$ is the family

the $\sigma$-algebra generated by the random variables $X_s$ with $s\le t$. This is a filtration: for $s\le t$ the generating family $\{X_u^{-1}(B):u\le s,\ B\ \text{Borel}\}$ of $\mathcal{F}^X_s$ is contained in $\mathcal{F}^X_t$, so the generated $\sigma$-algebra $\mathcal{F}^X_s$ is contained in $\mathcal{F}^X_t$ by minimality. The process $X$ is adapted to its natural filtration, since each generator $X_t^{-1}(B)$ lies in $\mathcal{F}^X_t$; and the natural filtration is the smallest filtration to which $X$ is adapted: if $X$ is adapted to a filtration $(\mathcal{F}_t)_{t\ge0}$, then for every $u\le t$ and Borel $B$ we have $X_u^{-1}(B)\in\mathcal{F}_u\subseteq\mathcal{F}_t$, so $\mathcal{F}_t$ contains the generating family of $\mathcal{F}^X_t$ and hence contains $\mathcal{F}^X_t$ by minimality.