Definition:Adapted Stochastic Process/Continuous Time

Definition
Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {X_t}_{t \ge 0}$ be a $\hointr 0 \infty$-indexed family of real-valued random variables.

We say that $\sequence {X_t}_{t \ge 0}$ is an adapted stochastic process :


 * $X_t$ is $\FF_t$-measurable for each $t \in \hointr 0 \infty$.