Definition:Stopped Process
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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 stopped process $\sequence {X_n^T}_{n \ge 0}$ by:
- $\map {X_n^T} \omega = \map {X_{\map T \omega \wedge n} } \omega$
for each $\omega \in \Omega$, where $\wedge$ is the pointwise minimum.
We write:
- $X_n^T = X_{T \wedge n}$
Also see
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $10.9$: Stopped supermartingales are supermartingales