Adapted process

Remarks

Every process is adapted to its natural filtration $\mathcal{F}_t^X = \sigma(X_s : s \le t)$, and this is the smallest filtration to which it is adapted.

Example: let $Y_1, Y_2, \dots$ be random variables with natural filtration $\mathcal{F}_n = \sigma(Y_1, \dots, Y_n)$. The partial sums $S_n = Y_1 + \cdots + Y_n$ form an adapted process, since $S_n$ is a measurable function of $Y_1, \dots, Y_n$.

Non-example: let the $Y_k$ be independent fair signs, equal to $1$ or $-1$, with the same filtration, and let $X_n = Y_{n+1}$. Then $X_n$ is independent of $\mathcal{F}_n$ and not almost surely constant, so it is not $\mathcal{F}_n$-measurable; the process $X$ looks one step into the future.

In continuous time a stronger notion is often needed: $X$ is progressively measurable if, for every $t$, the map $(s, \omega) \mapsto X_s(\omega)$ on $[0, t] \times \Omega$ is measurable with respect to $\mathcal{B}([0, t]) \otimes \mathcal{F}_t$. An adapted process with right-continuous or left-continuous paths is progressively measurable.