Properties of a Stochastic Process: Adapted and Predictable

A stochastic process is a collection of random variables indexed by time, all built on one outcome set. A single draw produces a whole path instead of a single number. Three properties place it against the flow of information: measurable, adapted and predictable, in that order of strictness. Only the strictest of the three can describe a rule somebody could follow.

One sentence carries the whole subject: one draw produces one path. The shift from a random variable to a process is worth slowing down on. The moment a single draw returns a whole sequence of readings rather than one reading, a new question appears that a random variable never had to answer. The question is when. When does each of those readings become known? A number that arrives all at once has no such question attached to it. A path that unfolds does, at every one of its points, and the three properties named above are three answers to it.

The pieces are already in place. A probability space, a set of events that stands for what is known, an average taken conditional on what is known, and one outcome set carrying more than one measure: all of that is settled elsewhere. The index is the new element. Once a clock is attached to the random variables, everything about the timing of knowledge has to be said out loud.

How is a stochastic process different from a random variable?

Random Variable vs Stochastic Process, Drawn on the Same Object

A random variableA single rule that turns one outcome into one number. is one rule that turns one outcome into one number. Draw once, get a number. Draw again, get another number. A stochastic processA collection of random variables indexed by time, all on one outcome set. is a whole collection of such rules, one for each time on the clock, all of them reading the same single outcome. Draw once, and every rule in the collection reads that same outcome and returns its own number. The draw returns not a number but the entire sequence, and that sequence has a name of its own: the pathThe whole sequence of values produced by one single draw..

The everyday version runs like this. A weighing scale read once gives a number, and that reading depends on the moment of looking. The single reading is a random variable. A scale wired to a recorder that logs continuously through the day gives a trace, and one day of that recorder is one draw, not a thousand draws. Somebody who reads the trace at noon and at four is reading two points of one object, not two separate experiments.

Take the invented standard processThe single invented traded quantity this subject area runs on, written S with a time subscript., written S with a time subscript, starting at Rs 100/-, drifting at 8 per cent a year, carrying a volatility of 20 per cent a year, watched for one year. The published twelve step path of that 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/-. The twelve readings came from one draw, not from twelve. A second draw would not change the ninth reading and leave the rest alone. A second draw would replace the whole line with a different line.

What times is a process defined at, and does the list of dates matter?

The collection needs a set of times to run over, and that set has a name: the index setThe set of times the process is defined at, whether a list of dates or a continuous stretch.. The index set can be a list of dates, in which case the process is a sequence and the arithmetic is ordinary algebra. The index set can instead be a continuous stretch of time, in which case there is a reading at every instant, uncountably many of them, and the mathematics gets considerably more demanding. The choice is not cosmetic. Almost every difficulty in this subject area comes from the second case.

The standard process is defined on the continuous stretch: it has a reading at every instant of the year, not just at twelve moments. The twelve step path is a partition chosen for drawing, and the choice of twelve belongs to the illustration rather than to the process. Choose twenty four steps and there are twenty four readings of the same object. Choose one and there are two, the start and the end.

The kilometre stones on a road are the everyday version. The stones are placed every kilometre because somebody chose that spacing, and the road exists between them regardless. Reading only the stones establishes that the road passed through those points; it does not establish that the road was straight in between, and it certainly does not establish that the road exists only at the stones. A coarse partition hides one property of paths almost completely: quadratic variation, set out under that name. The trap starts here.

What does it mean for a process to be measurable?

Measurable Process: The Weakest Property, and the One Nobody Notices

Everything above the weakest of the three is meaningless without it, so start there. Saying a process is measurableCapable of being assigned probabilities, jointly in time and outcome. is saying that probabilities can be attached to statements about it at all. Not that the probabilities are known, not that they are easy to compute, only that the statements worth making about it are the kind of statements a probability can be attached to.

For a single random variable that condition is familiar: the set of outcomes where the reading exceeds Rs 100/- has to be an event, meaning it has to sit inside the collection of sets the measure is defined on. For a process there is a second axis, and this is the part that is genuinely new. Statements about a process mix time and outcome together. The time spent above Rs 100/- during the year is a statement that runs across a stretch of time and across the outcome set at once, and only joint measurability makes it the kind of statement that can carry a probability.

The everyday version is the difference between what a scale reads and what the object weighs. Measurability says the scale has a dial at all, that the pointer lands somewhere on it, and that a range on that dial corresponds to a set of situations that can be counted. Measurability is not a statement that the pointer is accurate. Measurability is the condition that makes accuracy a question worth asking.

Almost every process encountered in practice is measurable, and that is exactly why the property gets skipped. Naming measurability anyway serves one purpose: it makes plain which job the property does. The other two properties are about timing. Measurability is not about timing at all. Measurability asks whether the object is the sort of object probability applies to, and the answer has to be yes before any question about timing can even be asked.

What is the information the other two properties are measured against?

The remaining two properties are both about what is known when, so something has to stand for what is known at each time, and that object is the filtrationThe growing collection of what is known, one entry for each time., written as a script F with a time subscript. The filtration is a list with one entry per time, and the entry at each time holds every statement that has been settled by then. Information settled at the end of month three is still settled at the end of month four, so the entries only grow, and nothing more is needed from the filtration below.

The filtration is set out in full under its own name, including what happens when the list is generated by the process itself and what changes when it is not. The two properties below use only the definition just given.

What does it mean for a process to be adapted?

Adapted Process: Known by Its Own Time, and Never Earlier

A process is adaptedKnown by the time it happens, and never earlier. when the reading at each time is settled by that time. Line the process up against the list of entries and check, at every time, that the reading at that time is one of the things the entry already holds. The line-up, repeated at every time, is the whole test.

Two things about it surprise people. The first is how easy it is to satisfy. The entry at each time was built to contain the history of the process, so any process checked against the list of entries generated by that history is adapted automatically, by construction. If the standard process is watched and nothing else is watched, the reading at month six is by definition part of what is known at month six. Adaptedness is close to free when the information is the process itself. A property that cheap is a poor filter and a bad thing to be reassured by.

The second surprise is how easy it is to break by accident. A quantity built from the process can fail the test without anybody noticing, and the way it fails is always the same: somewhere in its construction sits a reading that arrives later. A running average that quietly centres each month using the months on both sides of it is not adapted. A monthly figure restated three months afterwards, and then filed under the original month, is not adapted. The arithmetic has no idea what a calendar is, so nothing in it complains.

Run the test on the published path. The reading Rs 111.08/- belongs to month six, and at month six it is known: somebody standing there can see it. So the standard process is adapted. Now ask the harder question, the one the next property is built on. Standing at the end of month five, looking at Rs 101.35/-, is Rs 111.08/- known? It is not. Nothing available at the end of month five produces it. The reading at month six is known at month six and unknown at month five, and that single instant of difference is the entire content of the next section.

What does predictable mean, and why is it stricter?

Predictable Process: Settled One Instant Before It Applies

A process is predictableKnown just before it happens, which is what a decision needs. when the value at each time is settled strictly before that time. On a list of dates the statement is exact and pleasingly blunt: the value carrying the label of month six has to be settled at the end of month five. On a continuous stretch of time the same idea takes more care to state. The standard route is through processes that are adapted and left continuous. In plainer words, the value at an instant is reached from strictly earlier information rather than being handed over at the instant itself.

Notice how narrow the word is. Predictable has no connection with forecasting, skill, or a value being foreseeable in any ordinary sense. Predictability is purely a statement about the label on a clock: the quantity carries the index of a period and was settled before that period opened. A quantity can be predictable and completely wrong about what happens next. Being wrong is allowed, and it is the normal case.

The lift is the example worth keeping. A lift knows which floor it is on: that reading is adapted, settled at the moment it applies. The button pressed is predictable: it has to be pressed before the lift moves, using only the floor indicator as it stands at that moment. Nothing about pressing the button makes it a good choice. The point is only that the choice was fixed before the movement it applies to, and a rule that allows the button to be pressed after seeing where the lift ended up is not a rule anybody can operate.

The three properties do not sit beside one another. The three sit inside one another. A value settled before its own time is certainly settled by its own time, so predictable implies adapted. A value that lines up against the entries is measurable to begin with, so adapted implies measurable. The arrows only run one way, so predictable is the narrowest of the three and measurable the widest, and the standard process itself sits in the middle ring rather than the innermost one.

Why does one instant decide what can be acted on?

Here is where the distinction pays for itself. Suppose a rule says: hold one unit of the standard process through any month in which the process is above Rs 100/-. The rule does not say which reading it means. Two rules are hiding inside it, and they differ by one index.

The first rule looks at last month's reading and decides how much to hold through this month. The first rule is predictable: the quantity labelled month six was settled at the end of month five. The second rule looks at this month's own reading and decides how much was held through this month. The second rule is adapted and not predictable, and it is not a rule at all: at the moment the decision has to be taken, the number it needs has not arrived yet. The fault is look-aheadUsing at one moment a fact that only becomes available later., and it is the most common defect in results that look too good.

Both versions produce a perfectly well defined number. Multiply each month's position by that month's change in the process, add the twelve products, and the sum is plain arithmetic with no error in it anywhere. The two sums differ by Rs 23.02/- on the very same path, and no step of either calculation is wrong. Only one of them describes something somebody could have done.

The control does one thing at the first notch and another at the rest. Moving from setting zero to setting one changes the total by Rs 23.02/-, and that step is the only structural one on the whole control: it is the step from a quantity that is merely adapted to a quantity that is predictable. Moving from setting one to setting two changes the total again, and that step means nothing whatsoever. One path is all there is, and one path cannot establish which of two lawful rules is better. The first notch is a statement about what is possible and the rest are noise, and a reader who cannot tell those two kinds of movement apart will read every backtest wrongly.

Which of the three properties does the standard process have?

Checking is a fixed three step routine, and each step asks a different question about timing. A failure at one step makes the later steps meaningless rather than false, so run the three in order.

1. Can probabilities be attached at all? Ask whether statements about the process, including statements that mix a stretch of time with a set of outcomes, are the kind of statement a probability can attach to. For the standard process the answer is yes: it is measurable, jointly in time and outcome.
   Passes. Nothing further can be asked until it does.
2. Is each value settled by its own time? Ask whether the reading at each time is one of the things known at that time. Watching the standard process and nothing else, the answer is yes by construction: the reading at month six is part of what is known at month six.
   Passes, and it passes almost for free, which is why it proves so little.
3. Is each value settled strictly before its own time? Ask whether the reading at each time was known one instant earlier. Standing at the end of month five with Rs 101.35/- in hand, nothing available produces Rs 111.08/-. So the answer is no.
   Fails, and it is supposed to fail. A process this one is not predictable.

The standard process is measurable, is adapted, and is not predictable, and that combination is the normal case rather than a defect. If a process were predictable, its next reading would be settled before it arrived, and there would be nothing random about it in any useful sense. The whole point of a process worth modelling is that step three fails on it.

Which raises the question the routine is really for. If the process itself is never predictable, what is? The answer is the rule, not the process. Positions, holdings, decisions, quantities chosen by somebody: those are the objects required to be predictable, and they are required to be predictable precisely because the process is not. The asymmetry is the design. Something unpredictable is being watched, and something predictable is being decided against it.

How does somebody checking another person's work use this?

The distinction stops being vocabulary here and becomes a working habit. Somebody handed a set of results, a research note or a spreadsheet has to decide what to look at first, and the timing index on every decision is the highest yielding place to start. The check costs almost nothing, needs no understanding of the method, and catches a class of defect that no amount of recomputation will surface.

The routine is three questions. First, for each quantity that represents a choice rather than an observation, which reading did it use? Second, when did that reading become available? Third, is there a gap between those two moments, and does the gap run the right way? Any decision whose input arrived at the same instant as the outcome it responds to, or later, is not a decision, and everything downstream of it is a description of the past rather than a rule for the future.

The tell that costs nothing to look for is a result with no bad periods in it. A rule settled before each period opens will have periods where it was wrong, because the whole reason it is settled early is that the future was unavailable. On the published path the predictable version carries a position through month nine and month nine falls from Rs 101.12/- to Rs 93.74/-, and that is what an honest rule looks like: a position taken for a stated reason that then went the other way. A sheet where nothing ever goes the other way is not a better rule. Such a sheet shows a rule that has been shown the answers, and the sensible reaction is to go and read the subscripts.

The same habit applies to how data itself is filed. A monthly figure that gets revised afterwards and then stored under its original month has quietly made every later decision look better than it was, and no line of the analysis has to be wrong for that to happen. Filing by the date the number arrived rather than the date it describes is a small habit of hygiene that removes an entire category of this problem before it starts.

The standard process is invented.
Educational material. Not advice on any investment, tax, budget or market position.