Definition:Stopped Process

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:


 * $\map {X_n^T} \omega = \map {X_{\min \set {\map T \omega, n} } } \omega$

for each $\omega \in \Omega$, where $\min$ is the pointwise minimum.

We write:


 * $X_n^T = X_{T \wedge n}$

Also see

 * Stopped Process is Adapted Stochastic Process