Stopping Time: A Decision Rule That Cannot See Ahead

A stopping time is a rule for picking a moment where, at every moment, it can already be said whether the rule has fired. The test is nothing more than that question, asked at each date: is the answer to has it happened yet settled by what is known by then. A rule that needs any part of the future fails, however sensible it sounds.

The whole definition borrows one word from the properties of a process. A process is adaptedKnown by the time it happens, and never earlier. when its value at each moment is known by that moment. A stopping time applies the same requirement to a decision instead of to a value: the yes or no of having stopped must be settled by the moment it refers to. Everything else in this guide is that one requirement, tested against rules that sound reasonable until the test is run on them.

What single test decides whether a rule is a stopping time?

Here is the test in the form it actually takes. Stand at a moment. One question: can I say right now, with only what has already happened, whether this rule has fired. If the answer is yes at every moment, the rule is a stopping time. If there is even one moment where answering would require peeking at what comes later, it is not.

Notice what the test leaves out. The test does not ask whether the rule is sensible, profitable, popular or easy to compute. Nor does it ask whether the rule fires often or rarely. The test asks one thing, about knowability, and asks it at every moment rather than at the end.

The everyday version is easier to feel than the formal one. Think of a queue outside a ration shop. The rule leave the queue the first time it drops below five people is a rule that can be obeyed. A person stands there, counts, and at every second knows whether they have left. The rule leave the queue at the moment it is shortest today is not a rule that can be obeyed. At four in the afternoon the queue is six people long and there is no way of knowing whether it will be four at six in the evening. Which moment was shortest is learned only after the shop shuts, and by then leaving is not a decision, it is a description.

The difference between those two rules is not a matter of degree, and nothing else about stopping times matters as much. The first can be followed. The second can only be written down afterwards and read as though it had been followed.

One point of notation trips people, so take it before the examples. The condition is written with has it fired by now rather than does it fire exactly now. The two are equivalent when time moves in steps. Time does move in steps on the twelve monthly readings used here, so either version reads correctly. In continuous time the by now version is the one that behaves, and that is why it is the one written down.

Which rules pass the test, and why does each of them pass?

The workhorse is the first time a level is reached. The rule is called a first passageThe first moment a process reaches a stated level, and the standard example of a rule that can be followed. rule, and almost every stopping time in this subject is one of these or a small variation on it. A first passage rule passes the test in one line: standing at any month, the readings already in hand either include one that reached the level or they do not. No later reading is consulted, so the answer is available immediately.

Work it on the locked path of the standard process. Over twelve months the standard process reads Rs 97.64/-, Rs 107.63/-, Rs 100.35/-, Rs 100.27/-, Rs 101.35/-, Rs 111.08/-, Rs 103.56/-, Rs 101.12/-, Rs 93.74/-, Rs 96.41/-, Rs 102.06/- and Rs 106.18/-. Take the rule stop the first time the process reaches Rs 110/- or more. Month one at Rs 97.64/- does not reach it, and that is known at month one. Months two to five do not reach it either. Month six reads Rs 111.08/-, the rule fires, and it fires on the day, not in hindsight.

A second rule that passes looks as though it should fail, and is the more interesting for it. Stop the first time the process has fallen a tenth from its highest reading so far. The phrase so far is doing all the work. The running maximumThe highest reading up to the present moment. Knowable as it goes, unlike the highest reading of a whole period. is the highest reading up to the present moment, and it can be updated as the readings arrive with nothing but a pencil, so at every month both the current reading and the highest so far are known, and the two can be compared.

Run it. Start the running high at the opening Rs 100/-. Month one reads Rs 97.64/-, so the high stays Rs 100/- and the trip point is Rs 90/-, untouched. Month two reads Rs 107.63/-, a new high, and the trip point moves to Rs 96.867/-. The trip point stays there through months three, four and five. Month six reads Rs 111.08/-, a new high, and the trip point moves to Rs 99.972/-. Month seven at Rs 103.56/- and month eight at Rs 101.12/- both hold above it. Month nine reads Rs 93.74/-, below Rs 99.972/-, and the rule fires.

The rule fires at month nine at Rs 93.74/-, a fall of 15.610371 per cent from the high of Rs 111.08/-, and it can be seen to fire on the day because the trip point was written down before the reading arrived. That last clause is the test passing. The trip point at month nine was fixed by month six, and nothing from months ten, eleven or twelve was needed to know that month nine broke it.

The two get run together, so be clear about what a first passage rule does and does not report. The rule reports when, and only when. The rule says nothing about how far above the level the process went, nothing about how long it stayed there, and nothing about where it finished. On the locked path the rule fires at month six at Rs 111.08/-, and the year ends at Rs 106.18/-, below the level that fired it. Somebody who reads the firing as a statement about the finish has read a rule about arrival as a rule about outcome, and those are different objects with different answers.

There is a third rule that passes and that people forget is a rule at all: stop at the end of the period, whatever has happened. The rule fires at month twelve on every path. Asked at any month whether it has fired, the answer is a plain no until month twelve and a plain yes at month twelve. A fixed date is a stopping time. The point sounds like a technicality and turns out to matter. The amendment below is built out of exactly it.

Which reasonable sounding rules fail, and why is that surprising?

Now the rules that do not pass, and the point is that none of them sounds absurd. Each one is a sentence somebody could say out loud in a meeting without anybody blinking. The test exists for exactly that reason: the fault is not audible in the wording, and only shows up when somebody stands at a moment and asks the question.

Rule: stop at the highest reading of the year. On the locked path that is month six at Rs 111.08/-. At month six the process reads Rs 111.08/-, a new high, and there is no way at all of telling whether months seven to twelve will beat it. In fact month six is known to have been the highest only once month twelve has been read. The rule needs the whole year, so at month six the question has it fired has no answer.

Rule: stop at the last month the process reads below Rs 100/-. On the locked path the months below Rs 100/- are one, nine and ten, so the rule picks month ten at Rs 96.41/-. Stand at month ten. Month ten reads Rs 96.41/-, below Rs 100/-, but whether it is the last such month depends on months eleven and twelve. Any rule with the word last in it is a rule about the future, and this is worth carrying as a reflex: first is a word that can be acted on and last is a word that can only be written down afterwards.

Rule: stop one month before the peak. On the locked path the peak is month six, so this picks month five at Rs 101.35/-. The rule is the worst of the three. It needs a fact later than the moment it names by a full month even in the best case, and later by six months here. The rule also sounds the most like a method.

The pair is the cleanest illustration of the difference, so set the two maxima side by side. The running maximum and the maximum of the whole period are both maxima, both of the same process, and one is usable in a rule while the other is not. The running one is a process in its own right: it has a value at every moment, that value is known at that moment, and it never falls. The period maximum is a single number that comes into existence at the end. The period maximum is not hard to compute. Before month twelve it simply does not yet have a value.

All six rules are worth seeing on one sheet. The column that decides the verdict has to be worked out separately for each of them. The dates and the readings come straight off the locked path. The last column is the only one carrying any judgement, and it is judgement of a very mechanical kind: the earliest moment at which the rule could have been evaluated.

Read the second and fourth columns together and the pattern is exact: every rule whose knowable column matches its firing column passes, and every rule where the two differ fails. That is not a coincidence or a rule of thumb, it is the definition restated as a table. The three failing rules also all become knowable at month twelve, the expected tell: a rule that consults the whole period cannot be settled before the period ends, whatever moment it eventually names.

How can two rules give the same date and only one be a decision?

The comparison below is the sharpest thing available, and worth slowing down for. Put the two rules on the locked path side by side.

Three of the four columns agree exactly. The rules pick the same month, they pick the same reading, and a sheet of paper carrying only the answer would not reveal which rule produced it. The fourth column is the whole difference, and it is the column nobody prints.

Both rules answer month six on this path, the first knew it at month six, and the second could not know it until month twelve, so only one of the two was a decision anybody could have taken at the time. The agreement is a property of this particular path, not of the two rules. Move to almost any other path and the two part company. The first fires whenever the level is first reached, perhaps at month two, perhaps never. The second always fires at whichever month happens to carry the high.

The temptation this sets up is obvious once it is named. A rule that quietly uses the future will tend to look better than one that does not, and on the occasions where the two agree there is no visible trace of the difference. The output of both is a date. A date has no field on it for when it was knowable.

Move the control and the thing to watch is not the marker sliding along the path. The marker does not slide. The marker sits at month six for every level from Rs 107.64/- up to Rs 111.08/-, then leaps back to month two the moment the level drops to Rs 107.63/-, and stays at month two all the way down to Rs 97.65/-. Lowering the level by one paisa, from Rs 107.64/- to Rs 107.63/-, moves the date the rule fires by four whole months.

The control is also the cleanest way to see how narrow the agreement between the two rules really is. Sweep the level across the whole range from Rs 90/- to Rs 115/- one paisa at a time and there are 2,501 settings. Of those, 765 fire at month one, 999 fire at month two, 392 never fire at all, and only 345 fire at month six, the one answer that coincides with the highest reading of the year. The two rules agree on 345 of 2,501 settings, a share of 0.137945, and disagree on all the rest. A level picked at random is far more likely to land somewhere the two part company than somewhere they match. The coincidence in the worked instance is a warning rather than reassurance.

The jumping is not a quirk of this path. Jumping is what a first passage rule does. The rule does not measure how far the process got; it names the first arrival, and arrivals are events, not quantities. Small changes in a level shift which arrival counts as the first, and the answer moves in whole steps.

What happens on a path where the rule never fires at all?

Raise the level to Rs 115/- and the locked path never reaches it. The highest reading of the year is Rs 111.08/-, so there is no month at which the rule fires, and the rule as written has nothing to say about that path. A path that never reaches the level is not an edge case to be waved away. The third clause of every usable rule covers it, and the third clause is the one people leave out.

A rule stated properly has three parts. State the condition. State that the rule fires at the first moment the condition holds. State what happens if it never holds. Miss the third and the rule is undefined on a whole set of paths, and the first thing that will happen downstream is arithmetic on a moment the rule never produced.

The standard patch is the horizon cut-offThe clause that stops a rule at the end of the period on paths where its condition is never met.: fire at the first moment the level is reached, or at the end of the period if it never is. Formally it is the smaller of the first passage moment and the horizon, and it works because a fixed date is itself a stopping time, so the patched rule is one too.

One related word sits next to this one and means something different, and is worth having. A rule is almost surely finiteThe rule fires eventually on every path except a collection of paths carrying no probability between them. when it fires eventually on every path except a set carrying no probability. Almost surely finite is a statement about the whole set of paths, and not the same as firing within a stated period. With the drift and volatility of the standard process, the level of Rs 110/- is reached eventually with probability one, given indefinite waiting. Within one year is a different question with a different answer.

Set a second computed figure beside it and the reason for insisting on first arrival gets much sharper. The chance the process finishes the year at or above Rs 110/- is 0.429931. The chance it touches Rs 110/- at some point during the year is 0.721036. The two numbers are far apart because reaching a level and holding it are different events, and a rule about first arrival is a rule about the first, never about the second. The locked path is exactly this case: it reached Rs 111.08/- at month six and finished at Rs 106.18/-, so the rule fired and the finish is below the level.

One caution about those two figures, and it is a genuine subtlety rather than a footnote. Both figures assume the process is watched continuously. The locked path is a table of twelve monthly readings, and a rule checked monthly can miss a crossing that happened and unwound between two readings. So the monitoring frequency is part of the rule, not part of the background, and a rule that does not state how often it is checked has not been fully written down.

Can new rules be built out of rules already in hand?

The amendment above is quietly an example of something more general, and seeing it yields four new rules for free. Whichever comes first of the level being reached and the horizon arriving is itself a rule, and it passes the test because both of its ingredients do. The pattern generalises. Most rules met in practice are built rather than stated from scratch, so the pattern is worth carrying.

Take two rules that both pass. Call the first the level rule, firing at month six on the locked path, and the second the fall rule, firing at month nine. Whichever comes first fires at month six. Whichever comes later fires at month nine. Both of those are stopping times, and the reason is the same in each case: standing at any month, whether each ingredient has fired is known, so whether the combination has fired is known too.

Shifting a rule in time is where it gets interesting. Take the level rule and delay it by one month: fire one month after the level is first reached. On the locked path that is month seven at Rs 103.56/-. The delayed rule passes. At month seven whether month six reached the level is known, so whether the delayed rule has fired is known. Waiting is free, in the sense that it never costs knowability.

Move the same rule one month earlier and it fails immediately. At month five nothing reveals what month six will read. On the locked path the level rule fires at month six, so one month earlier is month five at Rs 101.35/-, which is exactly the answer the one month before the peak rule gave. The match is not a coincidence: on this path the first crossing of Rs 110/- and the peak are the same month, so the two backward shifted rules land in the same place. The lesson is that shifting a rule forward is always safe and shifting it backward never is, and that asymmetry is the arrow of time showing up in the arithmetic.

The box above invites a mistake worth warning about. The combinations are safe only when the ingredients are. Whichever comes first of the level rule and the highest reading of the year is not a stopping time. One of the two ingredients is not, and no amount of combining repairs it. Build only out of parts that have already been through the test individually.

What is known at a moment that is itself random?

Here is a question that sounds like a trick and is not. The moment the rule fires is random: it is month six on the locked path and it would be month two on a path that got there faster. So what does the information available at that moment mean, when the moment itself changes from path to path?

The answer is that it is defined path by path, and that is the whole of it. Walk one path. The rule fires at some particular month on that path. The information at a stopping timeWhatever had accumulated by the moment the rule fired, worked out separately on each path. on that path is whatever had accumulated by that month. Walk a different path and the rule fires at a different month and the information is whatever had accumulated by that different month. Nothing is ambiguous. The question is never asked in the abstract.

The formal statement looks strange the first time and it says exactly this. A fact counts as known at the moment the rule fired if, for every date, the combination of that fact and the rule having fired by that date is settled by what is known by that date. The definition is written this way rather than as whatever is known at the moment because the moment is random, and a random moment has to be reduced to statements about fixed ones.

The practical reading is short. Whatever was visible at the moment the rule fired may be used; whatever arrived afterwards may not, and the cut lands in a different place on every path. The amount of information at a stopping time is therefore itself variable, easiest to see by running one rule at three levels.

How does somebody reviewing a method use this rather than writing one?

The test is more useful on somebody else's work than on one's own. On one's own work the memory of what was known when is still there. Reviewing a written method, it is not, and the test gives the reviewer something to do that needs almost no knowledge of the subject matter.

The routine is short. Take the rule as written and list every quantity it consults. For each one, ask when that quantity first had a value. Then compare that against the moment the rule claims to fire. Any quantity whose value arrives later than the moment it is used in is a look-ahead, and one is enough to sink the rule.

1. List the quantities the rule consults Write them out one by one rather than reading the rule as a sentence. A rule reads smoothly and a list does not, and the roughness is the point.
   The highest of the year, the peak, the last time, the eventual finish. Each is a separate entry.
2. Date each quantity For each entry, write down the earliest moment at which it had a value. The high so far has a value at every month. The high of the year has a value at month twelve.
   This is where the fault becomes visible, and it takes about a minute.
3. Compare against the moment the rule claims Set the dates beside the moment the rule fires. Every quantity must be dated at or before it.
   One entry dated later is enough. There is no partial credit here.
4. Check the third clause exists Ask how the rule behaves on a path where its condition never holds. If the answer is missing, the rule is undefined on part of the set of paths.
   A rule with no horizon cut-off will eventually be handed a path it has no answer for.

Run those four steps on the highest reading of the year and it fails at step three, on the first entry, in under a minute. That is the value of having a test rather than a feel for it: the check is mechanical, it does not require agreeing with anybody about whether the rule is a good idea, and it produces the same verdict whoever runs it.

There is a second thing to watch for, and it is subtler than a look-ahead because nothing in the wording gives it away. A rule may consult only quantities that are properly dated and still be stated without saying how often it is checked. Checked monthly, the level rule fires at month six on the locked path. Checked continuously over the same year, it might have fired earlier, on a crossing that happened and unwound between two monthly readings, and the twelve published readings simply do not record whether that occurred. Neither version is wrong. The wrong choice is a rule that does not say which it is. The two produce different dates and the difference is invisible in the output.

The third thing worth asking about is the horizon itself. A rule with a cut-off at month twelve and the same rule with a cut-off at month twenty-four are different rules, and they will disagree on exactly the paths where the condition holds somewhere in the second year. On the locked path a level of Rs 115/- is never reached inside the year, so the twelve month version returns the horizon and a longer version might return a genuine crossing. A mismatch there will look like a disagreement about the rule when it is a disagreement about the period. Anybody comparing two sets of results should confirm the horizons match before comparing anything else.

A household version of the same discipline may help it stick. Somebody looking back over a year of electricity meter readings proposes the rule switch the supply the month the reading was highest. The question to put is when that month could have been known. The answer is: after the twelfth reading. Then the proposal is not a rule for switching, it is a description of a year that has already happened, and the two are being confused because they are written in the same words.

Why does this have to be settled before anything is priced?

Because the moment a decision enters a calculation, the calculation quietly acquires the decision's information requirements, and the arithmetic gives no sign of it. Every rule that shows up later in this subject, whether it triggers, cancels, extends or ends something, has to be checked against this one test first. If it passes, the machinery of averages and expectations applies to it in the ordinary way. If it does not, then a quantity computed by averaging over what happens after the rule has fired is averaging over information the rule already used, and the answer means something other than what it claims to.

There is a second reason this comes first rather than later, and it is about where the work gets done. Checking a rule against the test costs a minute and needs no model at all: it needs the rule, a list of the quantities it consults, and the dates those quantities acquire values. Discovering the same fault after a model has been built around the rule costs a rebuild. The fault is not in any one line of the model; it is in what the model was allowed to see. Cheap checks belong at the front, and this is the cheapest check in the subject.

The habit worth carrying away is smaller than the mathematics. Whenever a rule names a moment, the thing to do is stand at the moment it names and ask what was visible from there. One question separates a decision from a description, and it does so before any number has been computed, the only point at which the separation is cheap.

The standard process and its four parameters are invented.
Educational material. Not advice on any investment, tax, budget or market position.