Definition:Stopped Process

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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:

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

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