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 a (continuous-time) adapted stochastic process if and only if:

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

Sources

• 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): Definition $3.3$