Definition:Adapted Stochastic Process at Stopping Time
Jump to navigation
Jump to search
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
Sources
- 2014: Achim Klenke: Probability Theory (2nd ed.) ... (previous) ... (next): Definition $9.22$