Why a Brownian Motion’s First Future Zero Is a Stopping Time
Summary
The document distinguishes the last time Brownian motion is zero before a fixed time from the first zero after that time. It explains the distinction through the information available as time passes: a stopping time must be identifiable from observations up to the current moment, without needing future information.
After the fixed time, the first zero can be recognized as soon as it occurs, because no zero after the fixed time could have occurred earlier. By contrast, while approaching the fixed time, one cannot know that a zero is the last one before it; another zero may occur before the fixed time arrives. This is an intuitive explanation of the stopping-time property rather than a formal proof using a filtration or measurability criteria. The topic is stochastic-process theory and has no direct trading method or empirical market evidence.
Key ideas
- A stopping time can be identified using information observed up to that time.
- The first Brownian zero after a fixed time is observable as soon as it occurs.
- A zero before the fixed time cannot be confirmed as the last one until the fixed time has passed.
- The explanation is intuitive and does not provide a formal filtration-based proof.
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# On first and last zeros before t in a Brownian Motion
# On first and last zeros before t in a Brownian Motion
Suppose we have the following random variables, given a fixed $t$ we define the last zero before $t$ and the first zero after $t$:
\begin{align*} \alpha_t &= \sup\left\{ s\leq t: B(s) = 0 \right\}\\ \beta_t &= \inf\left\{ s\geq t: B(s) = 0 \right\}\ \end{align*}
Why $\beta_t$ is a stopping time but $\alpha_t$ is not?
Given the intuitive definition of a stopping time it makes much more sense in my head that the result would be the other way around.
## Answer by Hans-Peter Schrei (score 4)
https://quant.stackexchange.com/a/74704
Intuitively speaking, you generally have an event for which you do not know when it occurs (the time of the occurrence of the event is random), but you do know that it will occur at some point in the future. The event is a stopping time if you do know at any point in time whether the event has occurred or not.
In your example, you have a fixed time $t$. The event $\alpha_t$ is now describing the last occurrence of $B(s)=0$ before $t$. However, as you approach $t$ you will not know at any point if the last $B(s)=0$ before $t$ has occurred as there always could be another one coming up.
On the other hand, with $\beta_t$ you are now moving away from $t$ and you are waiting until you observe the first $B(s)=0$. As soon as you observe it, you will know that this was indeed the first $B(s)=0$ after $t$, as there could not have been another one before it.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.