Definition:Measurable Stochastic Process

Definition

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.

Let $\map \BB {\hointr 0 \infty}$ be the Borel $\sigma$-algebra of $\hointr 0 \infty$.

Let $\Sigma \otimes \map \BB {\hointr 0 \infty}$ be the product $\sigma$-algebra of $\Sigma$ and $\map \BB {\hointr 0 \infty}$.

We say that $\sequence {X_t}_{t \ge 0}$ is a measurable stochastic process if and only if the map $X : \Omega \times \hointr 0 \infty$ defined by:

$\map X {\omega, t} = \map {X_t} \omega$

for each $\tuple {\omega, t} \in \Omega \times \hointr 0 \infty$ is $\Sigma \otimes \map \BB {\hointr 0 \infty}$-measurable.