Jump Processes: How Intensity and Jump Size Are Set

A jump process moves by sudden discontinuous steps rather than only by continuous drift. Two things specify it, and they are set separately. The intensity is how often arrivals come. The size distribution is how large a move is when one does. The intensity says nothing whatever about the size, and that separation is the whole point.

Everything here follows from a single surrender. Up to now every path in this reading order joined up: a pencil could trace the whole year without once lifting from the paper. A jump processA process that moves by sudden discontinuous steps, so that its path does not join up at the moments those steps happen. gives that up on purpose. At certain random moments the level is one number an instant before and a different number an instant after, and there is no instant in between at which it was anything else. Once that is accepted, two questions open up that were never separate questions before. A continuous path never had to answer either of them. How often does that happen? And when it does, how far does the level move? The two questions have two separate answers, set by two separate pieces of the specification, and nothing in the model ties one to the other. The separation of the two is the whole subject, and everything below either unpacks it or follows from it.

An everyday example is worth having in place before the notation arrives. A lift in a building reports only which floor it is on. Most of the time nothing has changed, so nothing is reported at all. Then a report arrives. Two entirely separate facts govern what is observed: how often reports come, and how many floors the lift moved when one did. Knowing that reports arrive about twice a day says precisely nothing about whether the lift went up one floor or fell fourteen. The second fact is needed, and no amount of staring at the first will produce it.

What does a jump add that a continuous path cannot produce?

A jump produces a move that skips. The one word skips is the honest answer, and it is doing all the work. Continuity means that a continuous path travelling from Rs 111/- to Rs 90/- has to be at Rs 100/- at some instant on the way. A jump of the same size is at Rs 111/- and then at Rs 90/-, and there is no instant at which it was at Rs 100/-. The value was crossed without ever being attained.

The distinction sounds like a technicality and is not. Every rule that watches for a level, every trigger, every barrier, every stop, is a rule about attainment. A continuous path cannot get past a level without touching it, so a rule that watches a level cannot be evaded. A path with a discontinuityA moment at which the path does not join up, so the value an instant before and the value an instant after are different and nothing lies between them. can get past a level without ever being at it, and every rule built on the earlier assumption quietly changes meaning. Adding jumps does not make a model slightly rougher; it removes the guarantee that levels are attained, and that guarantee was load bearing.

One occurrence of a jump is called an arrivalOne occurrence of a jump. The word comes from counting: arrivals are what the counting process counts., and the word is worth adopting early because it keeps the two specifications apart. Arrivals are counted. Sizes are measured. Counting and measuring are different verbs applied to different objects, and confusing the two is the commonest error made about jumps.

What does jump intensity fix, and what does it leave completely open?

The jump intensityThe average number of arrivals per unit of time. Written lambda. It governs counting and nothing else. fixes how often, and it leaves everything about how large completely open. The intensity is a single positive number, written lambda, and it is read as an average count per unit of time. Set to 0.5 a year, it says that arrivals come at an average rate of half of one per year. The intensity says nothing else at all.

From that one number the entire counting behaviour follows. The chance of seeing exactly nought, one, two or three arrivals inside a year is fully determined once lambda is fixed, and there is no freedom left in it. The appeal of the intensity is exactly that: a single figure settles a whole distribution, and a single figure is easy to state, easy to compare and easy to put in a summary line. Hold on to that last observation. Ease of statement is what turns into the failure later.

With the locked figure in, the four chances come out. At an intensity of 0.5 a year over a one year horizon, the chance of no arrival at all is 0.606531, of exactly one is 0.303265, of exactly two is 0.075816 and of exactly three is 0.012636. Taking the first away from one gives the chance of at least one arrival, 0.393469. In roughly three years out of five nothing arrives at all, and that fact alone is what makes a low intensity feel so reassuring to read.

There is a second reading of the same number that is worth having, and it is the one that makes the intensity concrete. If arrivals come at an average rate of half of one a year, the average gap between one arrival and the next is two years exactly. The gap has a name, the waiting timeThe gap between one arrival and the next, or between now and the first arrival. It is itself random, with a distribution set by the intensity., and it is itself random rather than fixed. The waiting time has a distribution, and the distribution is settled by the intensity too.

Run both through the locked intensity. The average wait is one divided by 0.5, or two years exactly. The median wait is 0.693147 divided by 0.5, or 1.386294 years. The average and the median are two different numbers describing the same waiting time, and the gap between them is not a rounding artefact. Waiting times are skewed, so the typical wait is meaningfully shorter than the average wait, and a reader who treats the average as the typical case has already misread the intensity before ever reaching the size.

Why is jump size a distribution rather than a single number?

Because arrivals are not all the same size, and a model that said they were would be making a claim nobody believes. The jump sizeHow large a move is when an arrival happens, given as a distribution rather than as a single number, because arrivals differ from one another. is therefore given as a shape: a distribution, with its own parameters, entirely separate from the intensity. Fixing the intensity says nothing about this shape. Fixing this shape says nothing about the intensity.

The size is almost always given as a proportional jumpA jump applied as a multiple of the level rather than as a fixed amount, so a five per cent fall is a five per cent fall whether the level is Rs 100/- or Rs 1,000/-. rather than as an amount in rupees, and there is a plain reason for that. A jump specified as a fall of Rs 20/- means something completely different at Rs 100/- and at Rs 1,000/-, and at Rs 15/- it would take the level negative. A jump specified as a multiplier behaves the same way at any level and can never take a positive level below zero. So the size is a multiplier, and the natural way to describe a multiplier is through its logarithm.

Put the locked figures in. The logarithm of the multiplier has an average of minus 0.05 and a standard deviation of 0.10. The middle multiplier, the one half of all arrivals land below, is the exponential of minus 0.05, or 0.951229. On a level of Rs 100/- that is a fall to Rs 95.12/-. The average multiplier is not that number. The average multiplier is the exponential of minus 0.05 plus half of 0.01, giving 0.955997, or Rs 95.60/-. Averaging a multiplier and exponentiating the average of its logarithm are two different operations, and here they differ by 0.004768, or Rs 0.48/- on a Rs 100/- level.

The spread is what makes this a distribution rather than a figure. Run out the quantiles and the range is wide: the fifth percentile multiplier is 0.806957, a fall to Rs 80.70/-, and the ninety fifth is 1.121296, a rise to Rs 112.13/-. Note that last one. The average logarithm is negative, so the typical arrival is downward. Even so, the specification allows upward arrivals, and the chance an arrival moves the level up is 0.308538. Roughly three arrivals in ten are increases. The upward share is a fact about the size and could not have been recovered from the intensity by any route.

Why are the two set separately rather than rolled into one figure?

Because they are separate facts about the world, they are estimated by different means, they are argued about by different people, and rolling them together would destroy information that the model needs. The separation is the organising idea here, so it is worth taking slowly.

Start with what the separation means formally. The counting and the sizes are independent of each other. Independence here is a precise claim rather than a loose one. The chance of a particular count and the chances of particular sizes multiply. Nothing about the count shifts the size distribution and nothing about the sizes shifts the count distribution. Independence is not an approximation made for convenience. Independence is written into the specification as a condition, and any model that wants the two linked has to say so and pay for it in extra parameters.

Now the practical half, and it matters more. The two are estimated by different means. Counting how often something happened is a question about frequency, and frequency is the easier of the two to pin down because every quiet stretch is evidence: a decade with two arrivals is real information about the intensity. Measuring how large arrivals are is a question about magnitude, and magnitude is far harder. The whole point of the specification is the rare large move, and rare large moves are, by construction, the ones observed least often. The intensity is estimated from many observations of nothing happening. The size distribution is estimated from a handful of observations of something happening, and the second is always the weaker of the two.

The asymmetry in the evidence is why the two are argued about differently. A disagreement about the intensity is a disagreement about a count, and counts can be compared. A disagreement about the size distribution is a disagreement about the shape of a tail, and two people can hold wildly different views of a tail without either being contradicted by anything on record. Fold the two into one number and that whole argument disappears from view without being settled. For a quantity nobody is confident about, that is exactly the wrong outcome.

What do the two specifications say about one year of the standard process?

Now both halves on the same year, with the result read off. The standard process starts the year at Rs 100/-, and here it is carried by arrivals alone so that the two specifications can be seen without anything else moving. The intensity is 0.5 a year. The size shape has a logarithm averaging minus 0.05 with a spread of 0.10. Nothing else is set, and nothing else needs to be.

The formula shows the separation in the notation itself. The upper limit of the product comes from one specification. The thing being multiplied comes from the other. Changing the intensity changes the number of factors while every factor stays drawn from the same shape. Changing the size shape changes every factor while the number of them stays the same. The two specifications occupy two different positions in the same expression, and neither can be inferred from the other.

Two readings of that table are worth making explicitly. The first: in a clear majority of years the standard process is untouched by this mechanism entirely, finishing at Rs 100.00/- because nothing arrived. The second: the right hand column was computed from the size shape and the left hand column from the intensity, and the two columns were produced by two calculations that share no input. With the right hand column covered, it cannot be reconstructed from what remains. With the left covered, that column cannot be reconstructed either.

One caution on the right hand column. The column is an illustration rather than a promise, and it shows where a middle sized arrival lands, compounding the multiplier 0.951229 once, twice and three times. A real year with two arrivals would compound two draws from the shape rather than two copies of its middle, and those two draws could be anything the shape allows, including one large fall and one rise. The column is there to show that the size half exists and is doing work, not to summarise it.

What breaks in a model that assumed the path never jumped?

Three things break, and they break in order, each one because of the one before it. The temptation is to think of jumps as an addition. What jumps actually do is withdraw permissions, and the order of the three is worth walking through slowly.

The first is continuity itself, and it fails outright rather than partially. A path with arrivals is not a continuous path with a rough patch; it is a path that is discontinuous at certain moments and continuous everywhere else, and the definition of continuity is not the kind of thing that survives a single exception. Everything that was licensed by continuity has to be re-derived or dropped.

The second follows immediately. Continuity was what forced a small step in time to produce a small move in level. Removing it removes that implication: an interval of one second can contain an arrival, and an arrival is not small. A great deal of practical reasoning silently assumes that watching more often means seeing smaller moves, and losing that guarantee is more disruptive than it sounds. On a path with arrivals, watching more often does not shrink the moves observed. Watching more often only shrinks the chance of seeing a move at all.

The third is the one this reading order has been building toward. On a continuous path the sum of squared changes settled on the elapsed time. On a path with arrivals it does not, and the reason is exact rather than approximate.