Definition:Adapted Stochastic Process at Stopping Time

Definition

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

Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.

Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

We define the random variable $X_T$ by:

$\map {X_T} \omega = \map {X_{\map T \omega} } \omega \map {\chi_{\set {\omega \in \Omega : \map T \omega < \infty} } } \omega$

That is:

$\map {X_T} \omega = \begin{cases} \map {X_{\map T \omega} } \omega & : \map T \omega < \infty \\

0 &: \text{otherwise} \end{cases}$

Also see

• Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra

Sources

• 2014: Achim Klenke: Probability Theory (2nd ed.) ... (previous) ... (next): Definition $9.22$