Brownian Motion and Its Properties: The Standard Random Path

Brownian motion is the process fixed by four properties: it starts at zero, its changes over separate stretches of time are independent, the distribution of a change depends only on how long the stretch is, and its paths never jump. Nothing else is assumed. Everything else follows, including the rule that a change has a spread equal to the square root of the elapsed time.

The four properties are a specification rather than a description. A description says what something looks like and leaves room for other things that look similar. A specification pins down one object and refuses everything else. The four properties do the second thing: they admit exactly one process, and every further fact about Brownian motion is a consequence of them rather than an extra assumption bolted on afterwards. The tightness of that specification is why the list is worth memorising in the order given, and why the rest of this subject can lean so hard on a single rule about square roots.

What four properties define Brownian motion?

Take them one at a time. The process starts at zero. Over any two stretches of time that do not overlap, the two incrementsThe change in the path from one moment to a later one, which is the later value minus the earlier one. are independent. The distribution of an increment depends on the length of that stretch and on nothing else, and that distribution is normal with an average of nil and a variance equal to the length. And the path is continuous, meaning it can be traced without lifting a pen off the paper.

Notice how little is being claimed and how much it forces. There is no parameter to choose. There is no volatility to set, no speed to pick and no shape to assume. Once those four lines are written down all the freedom is used up, and anybody anywhere writing down the same four lines is talking about the same object. The version described here, starting at zero with a variance equal to the elapsed time, is what is meant by standard Brownian motionThe version that begins at zero and whose variance over any stretch equals the length of that stretch, with no scaling factor attached., and everything else in this subject is built by scaling and shifting it.

What do independent increments actually rule out?

Independent Increments

Independent incrementsChanges measured over stretches of time that do not overlap carry no information about one another, in either direction. is a much stronger condition than it sounds. The everyday reading is something like the process does not remember, and that reading is loose enough to cause real damage later. Independence actually says that being handed the change over one stretch, and told nothing else, changes the view of the change over a separate stretch by exactly nothing.

Work out what that forbids. Independence forbids momentum, where a rise over one month makes a rise over the next month more likely. Independence forbids reversal in the increments, where a rise over one month makes a fall over the next more likely. And it forbids clustering of magnitude, where a large move over one month makes the next month's move larger in size whichever way it points. Clustering is the exclusion people forget. It does not require the direction to be predictable. The loose reading of independence lets it through, and the actual reading does not.

Consider a rain gauge emptied every hour. Independence would say that a heavy hour says nothing about the next hour, neither that more rain is coming nor that a dry spell is due nor that the next hour will be more variable. Real rainfall is nothing like that. Independence is a strong assumption, not a throwaway one.

The last row of that picture carries the point, and it arrives early because it is worth arriving early. Independence constrains the increments. Independence says not one word about the level. The level is simply the running total of every increment so far, and the level is carried forward in full.

The distinction between independence and zero correlation is worth a moment. Zero correlation is a statement about one number, the average of the product. Independence is a statement about every question at once. A process can have perfectly uncorrelated increments and still have obvious structure between them, so independence rules out things that a correlation check would pass without comment. Clustering of magnitude is the standard instance: the increments can average out to no correlation while the sizes track each other closely.

What do stationary increments say, and what do they not say?

Stationary Increments

Stationary incrementsThe distribution of a change depends on how long the stretch is and not on when the stretch begins. is the shortest of the four properties to state and the easiest to over-read. Stationarity says that the distribution of an increment depends on the length of the stretch and not on where the stretch sits on the calendar. Both stretches are three months long, so an increment running from month two to month five has the same distribution as one running from month eight to month eleven.

Stationarity does not say that the two increments are equal, or close, or even similar. The two increments are different random quantities drawn from the same distribution, in the same way that two throws of the same die share a distribution and rarely share a face. Stationarity is a statement about the rule, never about the realisation, and confusing the two produces the belief that a stationary process should look repetitive. It should not. A stationary process should look different every time while obeying the same rule every time.

There is a household version of this that lands quickly. A weighing scale at a sweets counter has some error in it. Stationarity of that error means the size of the mistake it makes does not depend on whether the weighing happens at ten in the morning or at six in the evening. The size of the mistake depends on how much is being weighed. The scale is not promising to make the same mistake twice; it is promising that the rule generating the mistake does not shift with the clock.

Why is the standard deviation of an increment the square root of the time step?

The square root rule is the one fact everything else in the subject leans on. Deriving the rule beats accepting it. And the rule is not a fifth assumption. The rule falls straight out of the two properties already stated, in two short moves.

The first move. Take a stretch of length \(h_1\) followed immediately by a stretch of length \(h_2\). The middle value cancels in the addition, so the change over the whole span is the change over the first stretch plus the change over the second. The two stretches do not overlap, so they are independent, and the variance of a sum of independent quantities is the sum of their variances.

The second move is one line of algebra and one line of consequence. The spread is the square root of the variance, by definition, so if the variance is the length then the spread is the square root of the length. Variances add across time and spreads do not, and that single asymmetry is the whole content of square root scalingThe rule that the spread of a change grows with the square root of the length of the stretch it is measured over..

The everyday version of this is the most useful form of it. Suppose the sweets scale makes an independent error on each weighing. Weighing the same box four times and averaging the four readings halves the error in that average rather than quartering it. Everybody who has ever repeated a measurement to improve it has met square root scaling without naming it. Four times the effort buys twice the precision, and it is the same arithmetic on both sides: independent quantities add their variances, and the spread is a square root away.

What does continuity rule out, and what does it not bring with it?

Continuity is the fourth property, and it is the one most often assumed to be doing more work than it is. A continuous pathA path with no jumps, so that it passes through every value between any two values it takes. can be traced without lifting the pen. Continuity rules out jumps, and it rules out the path skipping over a level on its way past. If the path is at minus 0.2 at one moment and at plus 0.3 later, it visited every value in between at some point along the way.

Continuity does not bring smoothness with it. A smooth path would have a well defined direction at every point, a slope measurable by zooming in until the curve looked like a straight line. The same construction that produced the wiggle over a year produces an equally violent wiggle inside any month, any day and any minute of it. A Brownian path never looks like a straight line however far the zoom goes. Continuity and smoothness are two separate properties, and Brownian motion has the first and not the second.

The everyday version is a coastline. Paced out with a kilometre rule it has one length. Paced out with a metre rule, every bay that was cut across now has to be walked, so the length grows. Paced with a centimetre rule it grows again. The coast is continuous, in the sense that walking along it never requires a leap, and it has no direction at any point, in the sense that no rule short enough ever finds a straight stretch. Continuous everywhere and smooth nowhere is exactly the pair of properties in play here.

Is a Wiener process the same thing as Brownian motion?

Wiener Process

Yes. Same object, second name. A Wiener processA second name for the same process, after the mathematician who built the construction showing it exists. is Brownian motion, and the four properties above are its entire content under either name. The two names survive because they come from two histories. The physical name comes from the observation of pollen grains jittering in water, described in the eighteen twenties and explained in nineteen hundred and five. The mathematical name comes from Norbert Wiener, who in the nineteen twenties constructed the object rigorously and proved that a process with those four properties actually exists.

Writing down four conditions is not the same as showing that something satisfies them. The four properties are demanding enough that a process meeting all of them might have failed to exist. Wiener is credited for showing that one does. The notation here follows the mathematical name: the process is written with the letter W, which is the convention almost everywhere in this subject.

There is no third object hiding behind the two names, and no shade of difference between them worth learning. Where both appear in one document, they mean one thing. Some writers reserve the physical name for the process observed in a fluid and the mathematical name for the abstract object, but nothing in the mathematics turns on the distinction.

How is a teaching example built so that it reproduces exactly?

How to Build a Brownian-Motion Teaching Example

The practical problem is this. Every guide in this subject that draws a path needs a path to draw, and a path drawn fresh each time is useless for teaching. Drawn fresh, the reader who checks the arithmetic on Tuesday sees different numbers from the reader who checks it on Wednesday. The worked example is the only thing a careful reader can actually verify, and the worked example evaporates. So the path is built to a published recipe instead. Four steps, and the third is where it is won or lost.

1. Fix the horizon The span the path covers is decided and stated. One year here, which is the horizon used throughout this reading order.
   Everything downstream is measured in fractions of this, so it is written down first.
2. Fix the number of steps Choose how many equal stretches the horizon is cut into. Twelve here, so each step is one twelfth of a year and the square root of the step is 0.288675.
   Twelve gives a monthly reading, which is enough wiggle to see and few enough numbers to print.
3. Publish the driving values rather than drawing them Write down the twelve numbers that drive the path, in a table, in the open. Do not sample them. A published list is a published list on every device, on every reload, for every reader.
   This is the step that makes the whole thing reproduce. The other three are arithmetic.
4. Multiply each driving value by the square root of the step, and add them up Each driving value times 0.288675 gives that step's increment. The level at any month is the running total of the increments up to it.
   The square root, not the step itself. Using the step gives a path with the wrong spread entirely.

A recipe published with exact quantities is the difference between a dish anybody can reproduce and one that turns out differently in every kitchen. The published list is the whole trick, and it is not a compromise. Choosing the driving values deliberately lets them carry properties worth teaching. A drawn set could never be relied on to have those properties.

What does the locked path look like once it is built?

Running that recipe gives the following. The twelve driving values, in order, are minus 0.5, then 1.6, then minus 1.3, minus 0.1, 0.1, 1.5, minus 1.3, minus 0.5, minus 1.4, 0.4, 0.9 and 0.6. Every guide in this reading order that draws a path draws this one, and every figure derived from it reconciles to the table below.

Two properties were built into that list on purpose, and both are worth stating as construction rather than as coincidence. The twelve values sum to exactly zero, so the path returns precisely to its starting point at the end of the year. Their squares sum to exactly 12.0, so the twelve squared increments add to 1.000000, the elapsed time. A drawn set of twelve values would show neither of those exactly, and the fact that this one does is the direct payoff of step three of the recipe.

The path's lowest reading is minus 0.548483 at month nine and its highest is 0.375278 at month six, so it travels 0.923761 between its extremes on its way to finishing exactly where it began. Feed it into the standard processThe one invented traded quantity used as the worked instance throughout this reading order, starting at Rs 100/-, with a drift of 8 per cent a year and a volatility of 20 per cent a year. and it produces a run of monthly readings from Rs 97.64/- through a high of Rs 111.08/- at month six and a low of Rs 93.74/- at month nine, closing at Rs 106.18/-. The translation into a traded quantity is covered under geometric Brownian motion, and the driving values behind it belong here.

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

Almost nobody derives Brownian motion at work. Plenty of people are handed a model built on it and have to decide whether to believe the output, and for that job the four properties turn into four questions that need no access to the code.

The first question is about the scaling, and it catches the commonest implementation fault in the subject. Ask what the increments were multiplied by. If somebody has cut a year into two hundred and fifty steps and multiplied each driving value by the step, one over two hundred and fifty, rather than by its square root, every spread in the output is wrong by a factor of about 15.8. A model scaled by the step rather than its square root loses almost all its randomness as the partition is refined. The tell is that the answer barely moves when the number of steps is changed. Halve the step and the true spread should fall by a factor of 1.414214 and not by two.

The second question is about independence, and the third about stationarity. Was anything in the generator allowed to look back at the previous step? Does the size of a step depend on the calendar rather than only on its length? Both are ordinary modelling choices in other settings, and both quietly take the object outside the four properties, so anything downstream that relies on the square root rule stops holding.

The fourth question is the reproducibility one just answered. The fourth question asks whether the illustrative path in the document can be regenerated. If the answer is that it was drawn at run time and not recorded, then the worked example in that document cannot be checked by anybody, including its author. Losing the path is not a mathematical error, but it removes the only thing a reviewer could have verified independently, and in review terms that is close enough to the same cost.

One last point on scope, and it prevents a wrong inference. No jurisdiction sets the definition of a process. No exchange, regulator or accounting standard can amend the four properties, and no local convention narrows them. The four properties are the same four properties in every country and in every decade. The constancy is unusual enough in this subject area to be worth saying plainly.

The standard process and the twelve driving values behind the path drawn here are invented.
Educational material. Not advice on any investment, tax, budget or market position.