Category:Definitions/Progressive Stochastic Processes

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

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.

For each $s \in \hointr 0 \infty$, define the map $X^s : \Omega \times \closedint 0 s$ such that:

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

for each $\tuple {\omega, t} \in \Omega \times \closedint 0 s$.

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

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

We say that $\sequence {X_t}_{t \ge 0}$ is a progressive stochastic process if and only if $X^s$ is a $\Sigma \otimes \map \BB {\closedint 0 s}$-measurable function.