Stopping time, hitting time and other times

1. Formula
   Observation

   The slide text uses $W_t$ in the definition of a stopping time for a random process.

2. Diagram
   Observation

   The graph labels the fluctuating black path as $W_t$.

$W_t$

The random process under discussion; in the worked example it is the plotted fluctuating path.

$t \ge 0$

1. Formula
   Observation

   The definition line contains the symbol $\tau$ as the random time being defined.

2. Audio
   Observation

   The speaker says a stopping time is a random time tau.

$\tau$

A candidate stopping time for the process $W_t$.

Nonnegative random time.

1. Formula
   Observation

   The definition uses 'at time t' and the event {$\tau \le t$}.

2. Audio
   Observation

   The speaker repeatedly refers to knowing at time t whether to stop.

t

An observation time used to test whether the stopping decision is already determined by the past up to that time.

Fixed time parameter in the definition.

1. Formula
   Observation

   The slide states that {$\tau \le t$} is $F_t$ measurable.

2. Audio
   Observation

   The speaker reads '$F_t$ measurable'.

1. The underlying filtration is not explicitly defined on screen or in speech within this clip.

$\mathcal{F}_t$

The information available up to time t; the event {$\tau \le t$} must belong to this sigma-algebra for $\tau$ to be a stopping time.

Sigma-algebra associated with the process history up to time t.

1. Formula
   Observation

   The handwritten hitting-time definition uses level a in $W_t = a$.

2. Diagram
   Observation

   A red horizontal line is labeled a on the graph.

3. Audio
   Observation

   The speaker calls it 'some level a' and 'barrier a'.

a

A fixed level or barrier for the process; in the graph it is the horizontal threshold hit by $W_t$.

Real level shown around 6 on the vertical axis.

1. Formula
   Observation

   The handwritten definition is $T_a$ = inf{$t \ge 0$ : $W_t = a$}.

2. Audio
   Observation

   The speaker introduces this as the hitting time or first passage time.

$T_a$

The first time at which the process $W_t$ reaches level a.

Nonnegative extended random time.

1. Formula
   Observation

   The definition uses inf{$t \ge 0$ : $W_t = a$}.

2. Audio
   Observation

   The speaker paraphrases this as 'the first time ... that our random process hits level a'.

$\inf$

Infimum operator selecting the earliest time in the set of times when $W_t$ equals a.

Applied to a subset of [0,$\infty$).

1. Formula
   Observation

   Near the end of the clip, a second handwritten line appears: T^* = sup{$t \ge 0$ : $W_t = a$}.

1. Only the beginning of this line is visible before the clip ends; no explanation is given yet.

T^*

A later-defined time based on the supremum of times at which $W_t$ equals a.

Nonnegative extended random time.

1. Formula
   Observation

   Written as $τ_a$ = inf { $t \ge 0$ : $W_t = a$ }.

2. Audio
   Observation

   The lecturer discusses the hitting time / first passage time in this notation.

$\tau_a$

First hitting time (first passage time) of level a by the process W.

Nonnegative random time.

1. Formula
   Observation

   Appears in $τ_a$ = inf { $t \ge 0$ : $W_t = a$ }.

$\inf$

Infimum operator selecting the earliest time in the set.

1. Formula
   Observation

   Appears in $W_t$ inside the definitions of $τ_a$, $τ_b$, and τ_max.

2. Diagram
   Observation

   The plotted trajectory is labeled $W_t$.

$W_t$

The random process observed at time t.

1. Formula
   Observation

   Appears as the target level in $W_t = a$.

2. Diagram
   Observation

   A horizontal red line marks level a on the graph.

a

A fixed level of the process used to define hitting and last-hit times.

1. Formula
   Observation

   Slide text: 'A stopping time for a random process $W_t$ is a random time, $\tau$, such that we know at time t whether to stop, i.e. the event {$\tau \le t$} is $F_t$ measurable ({$\tau \le t$} depends only on the history of the process up to and including time t).'

2. Audio
   Observation

   The speaker reads and paraphrases the same definition, emphasizing that one does not need to look into the future.

1. The filtration ($F_t$) is named but not formally constructed in this clip.

A stopping time for a process $W_t$ is a random time $\tau$ such that, at each observation time t, the decision whether stopping has already occurred is determined by the process history up to and including t. Formally, the event {$\tau \le t$} must be $\mathcal{F}_t$-measurable.

1. Defined for a random process $W_t$.

2. The relevant information at time t is the history up to and including t.

3. The clip does not specify additional assumptions on the filtration.

1. Formula
   Observation

   Handwritten line: 'Hitting time / first passage time' followed by $T_a$ = inf{$t \ge 0$ : $W_t = a$}.

2. Audio
   Observation

   The speaker says the hitting time or first passage time is the first time for t greater or equal to zero that the random process hits level a.

3. Diagram
   Observation

   The graph shows a horizontal barrier a and a fluctuating path $W_t$ that reaches it.

The hitting time, also called first passage time, of level a is the earliest nonnegative time at which the process $W_t$ equals a. It is written as an infimum over all such times.

1. Level a is fixed.

2. The set is taken over times $t \ge 0$.

3. The clip does not discuss the case where the set is empty.

1. Definition of stopping time

1. Formula
   Observation

   Printed definition: 'A stopping time for a random process $W_t$ is a random time τ, such that we know at time t whether to stop, i.e. the event {$τ \le t$} is $F_t$ measurable ({$τ \le t$} depends only on the history of the process up to and including time t).'

2. Audio
   Observation

   The lecturer explains that one can decide whether to stop solely on the basis of information available up to and including time t.

A random time τ is a stopping time for a process if, at each time t, the decision whether τ has already occurred depends only on the history of the process up to and including t. Formally, the event {$τ \le t$} must be measurable with respect to the information available at time t.

1. τ is a random time for the process $W_t$.

2. Measurability is required for each observation time t.

3. Only information up to and including time t may be used.

1. Formula
   Observation

   Handwritten heading 'Hitting time / first passage time' followed by $τ_a$ = inf { $t \ge 0$ : $W_t = a$ }.

2. Audio
   Observation

   The lecturer treats this as an example of a time and asks whether it is a stopping time.

The hitting time of level a is the earliest nonnegative time at which the process W equals a. It is written as an infimum over the set of times satisfying $W_t = a$.

1. The process is observed for $t \ge 0$.

2. a is a fixed level.

1. Definition of stopping time

1. Formula
   Observation

   Under 'Other times:', the board shows $τ_b$ = sup { $t \ge 0$ : $W_t = a$ }.

2. Audio
   Observation

   The lecturer says 'let's define tau b as the last time that the process hits level a'.

1. The notation uses $τ_b$ but the defining equation still refers to level a; the video does not clarify this.

The lecturer introduces $τ_b$ as the last time the process hits level a, represented by taking the supremum over all times $t \ge 0$ such that $W_t = a$.

1. The process is observed for $t \ge 0$.

2. a is the same fixed level used in the hitting-time example.

1. Hitting time / first passage time

1. Formula
   Observation

   The board writes τ_max = max_{$0 \le t \le T$} $W_t$ and then asks 'Is this a stopping time?'

1. As written, the right-hand side is a maximum value of the process, not a time; the video does not resolve this notational issue.

The board presents another candidate object under 'Other times:' as τ_max = max_{$0 \le t \le T$} $W_t$ and asks whether it is a stopping time. The displayed formula itself denotes the maximum process value over the interval [0,T].

1. The maximum is taken over $0 \le t \le T$.

2. The video leaves the interpretation of this object unresolved within the clip.

1. Definition of stopping time

1. Formula
   Observation

   The formula $\tau_{\max} = \max_{0 \le t \le T} W_t$ is written on the board.

2. Audio
   Observation

   Speaker says: 'Max is just uh we stop when we reach the maximum of maximum of our process in the interval from zero to T.'

$\tau_{\max}$ is defined as the time point within the interval $[0, T]$ at which the stochastic process $W_t$ attains its maximum value.

1. Process $W_t$ is defined on $[0, T]$

1. Caption evidence
   Observation

   Text on screen: 'whether to stop, i.e. the event {$τ \le t$} is $F_t$ measurable ({$τ \le t$} depends only on the history of the process up to and including time t).'

2. Audio
   Observation

   Speaker argues that one cannot decide whether to stop using information available at time $t$ because future values are unknown.

A random time $\tau$ is a stopping time if the event $\{\tau \le t\}$ is $\mathcal{F}_t$-measurable, meaning the decision to stop by time $t$ depends only on the history of the process up to time $t$.

1. $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, P)$ is a filtered probability space

1. Audio
   Observation

   After asking whether $T_a$ is a stopping time, the speaker answers affirmatively and explains that one can decide to stop because the value of the process is observed.

2. Formula
   Observation

   The handwritten question 'Is this a stopping time?' is attached to $T_a$ = inf{$t \ge 0$ : $W_t = a$}.

3. Diagram
   Observation

   The graph marks the first contact point between $W_t$ and the horizontal level a.

1. The argument is presented informally on the example graph rather than as a full measure-theoretic proof.

For the process shown, the first hitting time $T_a = \inf$\{$t \ge 0$ : $W_t = a$\} is a stopping time because at any time t one can determine from the observed history whether the process has already hit level a.

1. There is a process $W_t$ and a fixed level a.

2. The decision is based on the history up to and including the current time.

For the observation times considered in the example, the event that $T_a$ has occurred by time t is determined by the path up to t.

1. Audio
   Observation

   The lecturer states that at time t we know whether to stop because we can observe the process and decide based only on information up to and including t.

2. Formula
   Observation

   The board asks 'Is this a stopping time?' next to $τ_a$ = inf { $t \ge 0$ : $W_t = a$ }.

For the hitting time $τ_a$ = inf{$t \ge 0$ : $W_t = a$}, the decision whether to stop by time t can be made using only the history of the process up to and including time t, so $τ_a$ satisfies the stopping-time criterion.

1. $W_t$ is the observed random process.

2. $τ_a$ is defined as the first time $t \ge 0$ with $W_t = a$.

3. The information available at time t is the history up to and including t.

For each observation time t.

1. Audio
   Observation

   The lecturer concludes 'tau b is not a stopping time because we don't have enough information available at time t here regarding whether this is the last time we actually hit level a.'

2. Formula
   Observation

   The board shows $τ_b$ = sup { $t \ge 0$ : $W_t = a$ } and asks whether it is a stopping time.

The time $τ_b$ = sup{$t \ge 0$ : $W_t = a$}, interpreted as the last time the process hits level a, is not a stopping time because deciding whether the current hit is the final one requires knowledge of the future evolution of the process.

1. $τ_b$ is defined as the supremum of times at which $W_t = a$.

2. The decision must be based only on information up to and including the current time t.

At a generic current time t when the process is at level a.

1. Audio
   Observation

   Speaker states: 'So is this a stopping time? Well clearly not... I can't decide where to stop using information available to me at time little t'.

2. Diagram
   Observation

   The graph illustrates that knowing the path up to time $t$ does not reveal if the current peak is the global maximum over $[0, T]$.

The random time $\tau_{\max} = \arg\max_{0 \le t \le T} W_t$ is not a stopping time with respect to the natural filtration of $W_t$.

1. $W_t$ is a non-trivial stochastic process (e.g., Brownian motion)

2. $T > 0$ is a fixed terminal time

For all $t < T$

1. Audio
   Observation

   The speaker asks whether $T_a$ is a stopping time, then reasons that if the process is at the hitting point, one can observe the value and therefore stop.

2. Diagram
   Observation

   The graph shows the path reaching the red barrier a for the first time.

1. This is an intuitive verification on the displayed sample path, not a formal proof for all processes and filtrations.

1. Expression

   $$$$
   Explanation

   Start from the definition of stopping time: at time t, one must already know whether the stopping event has happened.

   Justification

   Uses the definition ki-stopping-time-def.

   Shown in the video
2. Expression

   $$T_a = \inf\{t \ge 0 : W_t = a\}$$
   Explanation

   Rewrite the candidate time as the first time the process equals the level a.

   Justification

   Uses the definition ki-hitting-time-def.

   Shown in the video
3. Expression

   $$$$
   Explanation

   At a given time, inspect the observed path up to that time and check whether it has already touched level a.

   Justification

   The graph makes the path history visible up to the current point.

   Shown in the video
4. Expression

   $$$$
   Explanation

   If the path has reached a, the decision to stop is known immediately; no future values are needed.

   Justification

   This matches the informal explanation in the audio.

   Shown in the video

Therefore $T_a$ is treated as a stopping time in the example because the hitting event is detectable from the past up to the current time.

1. Audio
   Observation

   The lecturer explains that at time t we can observe the process and therefore decide whether to stop based only on information up to and including t.

2. Diagram
   Observation

   The graph shows the path $W_t$ and the level a, supporting the idea of checking whether the path has reached a by time t.

1. Expression

   $$$$
   Explanation

   Start from the definition $τ_a$ = inf{$t \ge 0$ : $W_t = a$}.

   Justification

   This is the displayed hitting-time definition.

   Shown in the video
2. Expression

   $$$$
   Explanation

   At any observation time t, one can inspect the realized path up to time t and determine whether the process has already hit level a.

   Justification

   The lecturer says we can observe the process and use information available up to and including time t.

   Shown in the video
3. Expression

   $$\{\tau_a \le t\}$$
   Explanation

   The event 'we have stopped by time t' is exactly the event that the process has hit a at some time no later than t.

   Justification

   This follows from the meaning of the infimum in the definition of $τ_a$.

   Derived from the video
4. Expression

   $$$$
   Explanation

   Since that event is determined by the history up to time t, $τ_a$ satisfies the stopping-time criterion.

   Justification

   The printed definition requires {$τ \le t$} to depend only on the history up to and including t.

   Shown in the video

$τ_a$ is a stopping time.

1. Audio
   Observation

   The lecturer says that staying at a point on the path, we cannot decide whether this is the last time we hit level a without knowing the future trajectory.

2. Diagram
   Observation

   The graph shows multiple crossings of the level a and a stick figure placed at one crossing, illustrating uncertainty about later behavior.

1. Expression

   $$$$
   Explanation

   Consider a current time t at which the process is at level a.

   Justification

   This matches the lecturer's setup of standing 'somewhere here' on the path.

   Shown in the video
2. Expression

   $$\tau_b = \sup\{t\ge 0: W_t = a\}$$
   Explanation

   To know whether the current time is $τ_b$, one must know whether there will be any later time at which W again equals a.

   Justification

   This is the meaning of the supremum over all hitting times of level a.

   Derived from the video
3. Expression

   $$$$
   Explanation

   That question cannot be answered from the history up to time t alone, because the future evolution of the process is unknown.

   Justification

   The lecturer explicitly says we do not have the benefit of hindsight and do not know what will happen in the future.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Therefore the event of having stopped by time t is not determined solely by information up to time t.

   Justification

   This contradicts the stopping-time requirement stated in the definition.

   Derived from the video

$τ_b$ is not a stopping time.

1. Audio
   Observation

   Step-by-step verbal reasoning provided by the speaker while pointing at the graph.

2. Diagram
   Observation

   Visual aid showing a stick figure at an intermediate peak and arrows indicating future uncertainty.

1. Expression

   $$$$
   Explanation

   Assume we are at some time $t < T$ and observe the path of $W$ up to $t$.

   Justification

   Setup for checking measurability condition.

   Shown in the video
2. Expression

   $$$$
   Explanation

   We want to know if the current observed maximum is the true maximum over $[0, T]$.

   Justification

   This corresponds to determining if $\tau_{\max} \le t$.

   Shown in the video
3. Expression

   $$$$
   Explanation

   However, the process $W$ may rise higher than the current maximum in the remaining time $(t, T]$.

   Justification

   Property of stochastic processes like Brownian motion having independent increments/future uncertainty.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Therefore, the event $\{\tau_{\max} \le t\}$ cannot be determined solely by the history $\mathcal{F}_t$.

   Justification

   Conclusion based on the inability to predict future maxima.

   Shown in the video
5. Expression

   $$$$
   Explanation

   Thus, $\tau_{\max}$ is not a stopping time.

   Justification

   Negation of the stopping time criterion.

   Shown in the video

$\tau_{\max}$ fails the adaptedness/measurability requirement for stopping times because deciding if the maximum has been reached requires knowledge of the future path up to $T$.

1. Diagram
   Observation

   A graph shows a fluctuating black path labeled $W_t$, a horizontal red line labeled a, and the first intersection point highlighted.

2. Audio
   Observation

   The speaker describes the process fluctuating up and down and hitting the barrier a for the first time.

1. The exact numerical coordinates of the first hit are not stated verbally; only the visual level a near 6 is clear.

Given a plotted random process $W_t$ and a horizontal level a, identify the first time the process reaches a and decide whether this defines a stopping time.

1. A fluctuating path $W_t$ is drawn on axes.

2. A horizontal barrier a is drawn across the graph.

3. The definition $T_a = \inf$\{$t \ge 0$ : $W_t = a$\} is written below.

Find the first hitting time of level a and test whether it is a stopping time.

1. Expression

   $$$$
   Explanation

   Read the graph as a sample path of the process over time.

   Justification

   The axes and label $W_t$ identify the curve as the process.

   Shown in the video
2. Expression

   $$a$$
   Explanation

   Identify the fixed horizontal level a as the target barrier.

   Justification

   The red line is labeled a in the diagram.

   Shown in the video
3. Expression

   $$T_a = \inf\{t \ge 0 : W_t = a\}$$
   Explanation

   Define the first time at which the path equals the barrier.

   Justification

   This is the handwritten formula shown on screen.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Locate the first intersection of the black path with the red line; that point represents $T_a$ on the graph.

   Justification

   The visual highlight marks the first contact with level a.

   Shown in the video
5. Expression

   $$$$
   Explanation

   Conclude that the decision to stop at that point is based only on the observed path up to that time.

   Justification

   The speaker explicitly says one can observe the value and therefore stop.

   Shown in the video

The first hitting time $T_a$ is the time of the first intersection between $W_t$ and the level a, and in this example it is presented as a stopping time.

Verified by comparing the graph's first contact point with the definition $T_a = \inf$\{$t \ge 0$ : $W_t = a$\} and the stopping-time criterion that the event be determined by the history up to the current time.

1. Formula
   Observation

   The board writes $τ_a$ = inf { $t \ge 0$ : $W_t = a$ } under 'Hitting time / first passage time'.

2. Audio
   Observation

   The lecturer uses this as the first example and argues it is a stopping time.

Given a random process $W_t$ and a fixed level a, define $τ_a$ = inf{$t \ge 0$ : $W_t = a$} and ask whether $τ_a$ is a stopping time.

1. A random process $W_t$.

2. A fixed level a.

3. The definition $τ_a$ = inf{$t \ge 0$ : $W_t = a$}.

Determine whether $τ_a$ satisfies the stopping-time criterion.

1. Expression

   $$\tau_a = \inf\{t\ge 0: W_t = a\}$$
   Explanation

   Define the first time at which the process reaches level a.

   Justification

   This is the displayed definition.

   Shown in the video
2. Expression

   $$$$
   Explanation

   At time t, observe the path up to t and check whether level a has already been reached.

   Justification

   The lecturer says the decision uses only information available up to and including time t.

   Shown in the video
3. Expression

   $$\{\tau_a \le t\}$$
   Explanation

   This event is determined by the history up to time t.

   Justification

   It records that some $s \le t$ satisfied $W_s = a$.

   Derived from the video

Yes, $τ_a$ is a stopping time.

The conclusion matches the printed stopping-time definition requiring {$τ \le t$} to depend only on the history up to time t.

1. Formula
   Observation

   The board writes $τ_b$ = sup { $t \ge 0$ : $W_t = a$ } under 'Other times:'.

2. Audio
   Observation

   The lecturer defines it verbally as the last time the process hits level a and concludes it is not a stopping time.

1. The notation $τ_b$ conflicts with the condition $W_t = a$; the clip does not explain the subscript choice.

Define $τ_b$ = sup{$t \ge 0$ : $W_t = a$}, interpreted as the last time the process hits level a, and ask whether $τ_b$ is a stopping time.

1. A random process $W_t$.

2. A fixed level a.

3. The definition $τ_b$ = sup{$t \ge 0$ : $W_t = a$}.

Determine whether $τ_b$ is a stopping time.

1. Expression

   $$\tau_b = \sup\{t\ge 0: W_t = a\}$$
   Explanation

   Define the candidate time as the supremum of all times at which the process equals a.

   Justification

   This is the displayed formula.

   Shown in the video
2. Expression

   $$$$
   Explanation

   At a current hitting time, decide whether this is the final hit of level a.

   Justification

   This is the decision problem emphasized by the lecturer.

   Shown in the video
3. Expression

   $$$$
   Explanation

   That decision requires knowledge of whether the process will hit a again in the future.

   Justification

   The lecturer states that without knowing the future trajectory we cannot decide.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Since the answer depends on future information, the stopping-time criterion fails.

   Justification

   The definition allows only information up to and including the current time t.

   Derived from the video

No, $τ_b$ is not a stopping time.

The reasoning directly applies the printed definition of stopping time and the lecturer's explicit conclusion.

1. Diagram
   Observation

   Top of the slide shows a hand holding a stopwatch above the title 'Stopping time' and the typed definition.

1. Stopwatch image

2. Title 'Stopping time'

3. Typed definition text

1. The cursor highlights parts of the definition while the speaker reads it.

1. The definition text remains on screen throughout this interval.

The stopwatch visually reinforces that a stopping time is about deciding when to stop based on elapsed observed time.

1. Diagram
   Observation

   The lower part of the slide shows a graph with a fluctuating black line labeled $W_t$ and a horizontal red line labeled a.

2. Animation
   Observation

   Around 122-128 seconds, a stick figure and arrow are drawn above the graph pointing toward the first intersection with the red line.

1. The precise x-coordinate of the first intersection is not labeled numerically.

1. Black path $W_t$

2. Red horizontal line a

3. Axes

4. Stick figure annotation

5. Arrow annotation

1. The view scrolls down to reveal the graph and the handwritten hitting-time formula.

2. A mark is placed at the first intersection of $W_t$ with level a.

3. An arrow and stick figure are added to emphasize the hitting moment.

1. The level a stays fixed as a horizontal barrier.

2. The path $W_t$ is the evolving object whose first contact with a is being studied.

The animation identifies the first passage time as the earliest time at which the observed path reaches the fixed barrier.

1. Diagram
   Observation

   A graph shows axes labeled [$W_t$] vertically and [t] horizontally, a jagged black path labeled $W_t$, a horizontal red line labeled a, and stick figures above the path with arrows pointing down to points on the trajectory.

2. Animation
   Observation

   Across the clip the view scrolls between the definition/graph area and the handwritten examples, and additional stick-figure annotations are visible near later times.

1. Exact numerical coordinates of the marked points are not stated in the audio and are only approximate from the drawing.

1. Vertical axis labeled [$W_t$].

2. Horizontal axis labeled [t].

3. Black jagged trajectory labeled $W_t$.

4. Horizontal red line labeled a.

5. Stick figures with downward arrows marking positions on the path.

1. The page scrolls between the typed definition and the handwritten examples.

2. The visual focus shifts from the general stopping-time definition to the specific formulas for $τ_a$, $τ_b$, and τ_max.

3. Additional stick-figure annotations emphasize different points on the trajectory during the discussion of last hitting time.

1. The red line remains fixed at level a.

2. The trajectory $W_t$ is the same plotted path throughout the graph views.

The diagram supports the verbal distinction between first hitting and last hitting: reaching level a can be detected from the past up to the current time, but identifying the final visit to level a would require seeing the entire future of the path.

1. Diagram
   Observation

   A hand-drawn graph of a jagged line ($W_t$) oscillating around a horizontal axis, intersecting a red horizontal line ($a$). Stick figures represent observers at different times.

1. Jagged black line representing $W_t$

2. Horizontal red line representing level $a$

3. Stick figures on the curve

4. Arrows indicating time progression and future possibilities

1. Focus shifts from the general definition to specific points on the curve to illustrate the concept of 'history' vs 'future'.

1. The underlying mathematical object being discussed is the sample path of the stochastic process.

The visual demonstrates that at any intermediate point (stick figure), one cannot know if the current height is the global maximum without seeing the rest of the path (arrows pointing right).

1. Audio
   Observation

   The speaker says: 'We don't need to look into the future to decide whether to stop or not.'

One might think a stopping rule can depend on what happens after the current time.

The video stresses that a stopping time must be decided using only the history up to and including the current time.

1. Audio
   Observation

   The lecturer repeatedly contrasts decisions based on information up to time t with decisions that would require looking into the future.

2. Formula
   Observation

   The stopping-time definition explicitly says {$τ \le t$} depends only on the history up to and including time t.

One may think any naturally defined random time is automatically a stopping time.

A stopping time must be decidable using only the history up to the current time. The last-hit time $τ_b$ fails because recognizing the final hit requires future information.

1. Formula
   Observation

   The board writes τ_max = max_{$0 \le t \le T$} $W_t$ and asks whether this is a stopping time.

1. This item is an analyst note about the displayed notation, not an explicit correction spoken in the clip.

The symbol τ_max suggests a random time, but the displayed right-hand side max_{$0 \le t \le T$} $W_t$ is a maximum value of the process rather than a time.

If the intended object were the time at which the maximum is attained, a more standard definition would be something like argmax_{$0 \le t \le T$} $W_t$ or the first time attaining the running maximum. The clip itself does not resolve this ambiguity.

1. Formula
   Observation

   Contrast between $\tau_a$ (hitting time) and $\tau_{\max}$ (time of maximum).

2. Audio
   Observation

   Speaker explicitly asks 'Is this a stopping time?' implying it's a common question/point of confusion compared to the previous examples.

Students might assume that since hitting a level $a$ ($\tau_a$) is a stopping time, finding the maximum ($\tau_{\max}$) is also a stopping time.

Hitting a fixed level $a$ allows you to stop immediately upon contact (past info suffices). Finding the maximum requires comparing the current value against all future values up to $T$ (future info needed).

1. Formula
   Observation

   The hitting-time definition is introduced directly beneath the stopping-time definition.

2. Audio
   Observation

   The speaker asks whether the hitting time is a stopping time and then answers yes.

The hitting time $T_a$ is introduced as a concrete example to which the abstract stopping-time definition is applied.

1. Audio
   Observation

   The speaker concludes that the first hitting time / first passage time is indeed a stopping time.

2. Formula
   Observation

   $T_a = \inf$\{$t \ge 0$ : $W_t = a$\} is checked against the stopping-time criterion.

1. This relation is derived from the video's example rather than stated as a general theorem in full formal language.

Within the clip, $T_a$ is presented as a specific instance of a stopping time: the first time the process reaches a fixed level.

1. Formula
   Observation

   The examples $τ_a$ and $τ_b$ are each followed by the question 'Is this a stopping time?' after the stopping-time definition is shown.

The hitting-time example is evaluated using the definition of stopping time.

1. Formula
   Observation

   The board asks whether $τ_b$ is a stopping time immediately after presenting the stopping-time definition.

The last-hit-time example is also classified by testing the stopping-time criterion.

1. Audio
   Observation

   The lecturer contrasts the first hitting time, which can be decided from the past, with the last hitting time, which cannot.

2. Formula
   Observation

   The board juxtaposes $τ_a$ = inf { $t \ge 0$ : $W_t = a$ } and $τ_b$ = sup { $t \ge 0$ : $W_t = a$ }.

Both concern visits to the same level a, but the infimum version yields a stopping time while the supremum version does not.

1. Audio
   Observation

   Direct application of the stopping time definition to the specific example of $\tau_{\max}$.

The definition of $\tau_{\max}$ is tested against the criterion for stopping times, resulting in a negative conclusion.

1. Formula
   Observation

   The slide gives the formal definition of stopping time.

1. Definition of stopping time

1. Audio
   Observation

   The speaker says one does not need to look into the future to decide whether to stop.

1. Definition of stopping time
2. Confusing stopping times with decisions that require future knowledge

1. Formula
   Observation

   The handwritten formula $T_a$ = inf{t >= 0 : $W_t = a$} is shown.

1. Definition of hitting time / first passage time

1. Audio
   Observation

   The speaker asks whether $T_a$ is a stopping time and then answers affirmatively.

1. Definition of hitting time / first passage time
2. The hitting time $T_a$ is a stopping time
3. Intuitive verification that $T_a$ satisfies the stopping-time condition

1. Diagram
   Observation

   The graph shows $W_t$ crossing the horizontal level a.

1. Graphical example of a hitting time
2. Graphical depiction of hitting level a

1. Formula
   Observation

   The printed definition of stopping time is visible on the board.

1. Definition of stopping time

1. Audio
   Observation

   The lecturer explains why the hitting time can be decided from information up to time t.

1. Hitting time / first passage time
2. The hitting time $τ_a$ is a stopping time
3. Why $τ_a$ is a stopping time

1. Audio
   Observation

   The lecturer says we cannot know whether the current hit is the last one without future information.

1. Last hit time
2. The last-hit time $τ_b$ is not a stopping time
3. Why $τ_b$ is not a stopping time

1. Formula
   Observation

   The board shows $τ_a$ with inf and $τ_b$ with sup over the same hitting condition $W_t = a$.

1. Hitting time / first passage time
2. Last hit time
3. s150-cr-first-vs-last

1. Formula
   Observation

   The board writes τ_max = max_{$0 \le t \le T$} $W_t$ and asks whether it is a stopping time.

1. The clip does not provide the answer or clarify the notation.

1. Displayed τ_max formula
2. Notational ambiguity in the displayed τ_max formula

1. Audio
   Observation

   The core question answered in the clip.

1. Definition of $\tau_{\max}$
2. $\tau_{\max}$ is not a stopping time
3. Proof sketch that $\tau_{\max}$ is not a stopping time