Stopping Time

Last updated Nov 1, 2022

Suppose ${\mathcal{B}{n} : n \in \mathbb{N} }$ is a Discrete-Time Filtration on $\Omega$. Then a Function $\nu : \Omega \to \bar{\mathbb{N}}$ is a Stopping Time if $$[\nu = n] \in \mathcal{B}{n}, \forall n \in \mathbb{N}$$

1. Strategy Corresponding to a Stopping Time
2. $\nu$ can be $\infty$. In that case, $\nu$ represents the case that we never stop.
3. Equivalent Conditions for being a Stopping Time
4. Stopping Times are Measureable with respect to the Borel Sigma Algebra on the Order Topology of the Extended Naturals

Observe that $$[\nu = \infty] = \bigcap\limits_{n \in \mathbb{N}} [\nu \neq n]^{C} = \left(\bigcup\limits_{n \in \mathbb{N}} [\nu = n]\right)^{C} \in \mathcal{B}\infty$$ since $[\nu = n] \in \mathcal{B}{n} \subset \mathcal{B}_\infty$ $\forall n \in \mathbb{N}$. Thus, our defining property can be extended to $\bar{\mathbb{N}}$.

1. Constant Function is a Stopping Time
2. Equivalent Conditions for being a Stopping Time
3. Extrema of Stopping Times are Stopping Times
4. Limits of Monotone Sequences of Stopping Times are Stopping Times
5. Sums of Stopping Times are Stopping Times

TODO - this could probably be reworked for general Stopping Times by allowing the image of the stopping time to be a Total Ordering, then applying Compactification on the Order Topology. For now, I’ll leave as a discrete construct. If need be, I can change the name of this to “Discrete Stopping Time” then rewire the dependent pages.

• Natural Numbers
• Extended Natural Numbers