Category:Definitions/Adapted Stochastic Processes

This category contains definitions related to Adapted Stochastic Processes.
Related results can be found in Category:Adapted Stochastic Processes.

Discrete Time

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \in \N}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \mathop \in \N}$ be a sequence of real-valued random variables.

We say that $\sequence {X_n}_{n \mathop \in \N}$ is an (discrete-time) adapted stochastic process if and only if:

$X_n$ is $\FF_n$-measurable random variable for each $n \in \N$.

Continuous Time

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 $t \in \hointr 0 \infty$.