Levy Processes: The General Class Behind Every Jump Model

A Levy process is any process starting at zero with independent increments whose distribution depends only on the length of the stretch, and with paths that have no worse than jump discontinuities. The list is that short, and it is the whole definition. Every such process splits into exactly three parts: a steady drift, a continuous random part and a jump part.

Several processes have now been met one at a time, each with its own specification and its own picture. A definition short enough to memorise in a minute covers all of them, and a result then says that anything satisfying it is built from exactly three ingredients and nothing else. The definition is easy. The prize is the completeness of the result that follows from it: three slots, and no fourth slot exists. An unfamiliar process can then be taken up, three questions asked of it, and every part of its behaviour traced to its source.

What defines a Levy process, and how short is the list?

The shape of the definition is the surprise before any of its content is. Start there. The general description of processes that jump might reasonably be expected to be long and technical, full of conditions guarding against pathologies. The description is four lines. Three of those four lines appear word for word in the specification of Brownian motion. Only the fourth is different, and it is different by being weaker rather than stronger.

Line one says the process starts at zero. Starting at zero is bookkeeping and costs nothing. Any process can be shifted so it starts there, and the movement is what is being described rather than the starting level. Line two says that the change over one stretch of time carries no news about the change over a separate stretch. Line three says the distribution of an incrementThe change in a process between two times, being the value at the later time less the value at the earlier one. depends on the length of the stretch and on nothing else, so a one month change looks the same in January as in September. Line four is where everything that follows lives.

Brownian motion answers line four with never jumps. The general class answers it with jumps are allowed, but nothing worse than a jump is. Worse than a jump sounds like a joke. It is not, and the phrase needs unpacking. A path is required to be right continuous with left limitsAt any instant the path takes the value it is about to hold going forward, and the value it held an instant earlier always exists. A break is therefore a clean step with a definite value either side., which means two things in plain terms. At any instant the path has already taken its new value, and an instant before the break there was a definite old value. So a break is a clean step from one number to another number. Two kinds of path are forbidden: one that oscillates infinitely fast near a moment, and one that has no value approaching a moment from the left. For either, there is no saying what jumped to what.

The everyday version comes from measuring rather than from anything financial. Consider pacing out a long wall with a metre rule, laying it end to end and writing down the running total. Every laying is the same operation, and the distribution of what gets added is the same each time. Line three holds. No laying is influenced by the last one. Line two holds. The tally started at nothing. Line one holds. And the running total either creeps up smoothly or steps by a whole rule length; what it never does is become undefined halfway along. The refusal to become undefined is line four, and stating it explicitly is what stops the mathematics from having to handle objects nobody wants.

Notice one omission. The definition never names a distribution. Brownian motion pins its increments to the normal law; the definition here leaves the shape of the increment open and constrains only how increments relate to one another and to the clock. The refusal to name a shape is exactly why the class is large. The definition constrains the relationships between increments and says almost nothing about any single increment. Such generality is what makes it worth setting out on its own.

What three parts does every one of them split into?

Here is the result the class exists for, and it carries the names of Levy and Ito. Take any process satisfying those four lines. The process can be written as the sum of a straight line, a scaled Brownian motion and a jump part, and those three are all there is. Not usually. Not for the well behaved cases. For every process in the class, without exception and without further conditions.

The first part is the driftThe steady, non-random part of the movement. The drift advances by the same amount in every stretch of equal length., and it draws a straight line advancing at a constant rate. The drift has no randomness in it at all: over any stretch of a month it contributes exactly one twelfth of its annual rate, every month, forever. The second part is a Brownian motion multiplied by a constant, and it supplies the continuous randomness. Its size is quoted as a variance rateThe variance the continuous part contributes over one year. Its square root is the volatility, so a variance rate of 0.04 a year corresponds to a volatility of 0.20 a year., the amount of variance it contributes per year. The third part collects the jumps, described by how often they arrive and by how large they are when they do.

Because the three are exactly determined by three pieces of information, a process in the class is completely described by a rate, a variance rate and a jump specification. The three quantities together are called the characteristic tripletThe three pieces of information that pin down a process in this class completely: the drift rate, the variance rate of the continuous part, and the specification of the jumps., and once the triplet has been written down there is nothing further to say about the process. Two processes with the same triplet are the same process. The decomposition is a complete inventory rather than a convenient framing, and that completeness is the entire content of the result.

One practical caveat belongs here rather than later. Where arrivals come at a finite rate, meaning only finitely many of them in any year, the jump part can be written as a plain running sum of the sizes that have arrived. Every process in this guide has an intensity quotable as a number of arrivals a year, and every one of them is therefore of that kind. Where a process is allowed infinitely many tiny jumps in any stretch, the jump part still exists but has to be written with a correcting term so the sum stays finite. The correcting term changes the bookkeeping and changes nothing about the count of three.

Which of the processes already met sit inside the class?

Abstraction earns its keep only when it places things already known. So the four processes running through this reading order each have their three slots filled in, one at a time. All four numbers below belong to the standard process and are fixed in advance rather than observed. Every reading is therefore reproducible.

Brownian motion first. Standard Brownian motion has no steady advance. Its drift rate is zero. Its continuous part has a variance rate of 1.0 a year. Standard means exactly that: over one year the variance of the change is one. Its jump slot is empty. One slot filled out of three, and it is the simplest member of the class precisely because the other two boxes are blank.

The standard process in logarithms next, and this is the row worth checking by hand. The standard process has a drift of 8 per cent a year and a volatility of 20 per cent a year, both invented and fixed. Its logarithm advances at the drift less half the variance rate: 0.08 less 0.02, giving 0.06 a year exactly. Its continuous part carries the variance rate 0.04, being 0.20 squared. Its jump slot is empty. Two slots filled.

The counting process at an intensityThe average number of arrivals per unit of time, quoted here as arrivals per year. The intensity is the only freedom a counting process has. of 0.5 arrivals a year is the mirror image of Brownian motion. Its drift rate is zero, its continuous variance rate is zero, and its jump slot alone is filled, with arrivals at 0.5 a year and every arrival of size exactly one. Between arrivals it does not move at all; when it moves it only jumps. One slot filled, and a different one.

The jump diffusion that closes this reading order fills all three. Drift 0.06 a year, continuous variance rate 0.04 a year, and arrivals at 0.5 a year whose proportional size has a logarithm averaging minus 0.05 with a standard deviation of 0.10, so a typical arrival moves the level downward. The jump diffusion carries the name of Merton, 1976, in the literature.

Read that table across and the general class stops being abstract. Four processes that look nothing like each other on paper turn out to be four different fillings of the same three boxes, and nothing in any row required a box the other rows did not have. Read down instead and a second reading appears: the drift column holds two zeros and two sixes, the continuous column holds a one, a zero and two identical variance rates, and the jump column holds two blanks and two intensities that agree.

Placing an unfamiliar process is therefore a three question routine rather than an act of judgement. Does it advance by a fixed amount in every stretch of equal length, whatever the randomness does. Does it wobble continuously between its breaks. Does it break at all. Three answers, three slots, and the process is placed.

Why does the same description work at every time scale?

Lines two and three have a consequence that deserves its own block. The consequence is why these processes are convenient to compute with, and why the same three numbers can be quoted at any frequency. Take a change over one year. Cut the year into twelve months. The twelve monthly changes are independent by line two, and they all have the same distribution by line three, and they add up to the annual change by arithmetic. So the annual change has been written as a sum of twelve independent pieces that are identical to one another.

Now notice that nothing chose twelve. The same argument works for two halves, for four quarters, for 365 days, for any number at all. A quantity that can be written as a sum of any number of identical independent pieces is called infinitely divisibleA quantity that can be split into any number of identical independent pieces that add back to it. Every increment of a process in this class has this property., and every increment of every process in this class has that property automatically. Infinite divisibility is not an extra assumption; it falls straight out of the two lines already accepted.

Work it on the standard process in logarithms, where the annual triplet is a drift of 0.06, a variance rate of 0.04 and no jumps. Split the year in two and each half carries a drift of 0.03 and a variance of 0.02. Split it in four and each quarter carries 0.015 and 0.01. Split it in twelve and each month carries a drift of 0.005 and a variance of 0.003333, whose square root is a standard deviation of 0.057735. The monthly standard deviation is also the locked volatility of 0.20 multiplied by the square root of one twelfth, being 0.288675. The two routes are the same calculation written twice and agree to six decimals.

The reason to care is practical rather than decorative. Because the three quantities all scale in proportion to the length of the stretch, a triplet quoted per year can be converted to any other frequency by one division and never by a rule of thumb. Anything that does not scale that way has broken one of the two lines, and the failure will show up as a triplet that refuses to stay the same when the reading frequency changes.

What does the split buy that studying each process alone does not?

The honest answer is not elegance. The gain is a short list of suspects. When a process in this class behaves unexpectedly, the decomposition establishes that the behaviour has exactly three possible sources, and they can be gone through one at a time before running any calculation at all.

Here is a concrete instance built entirely from the locked figures. The locked path is the twelve step path published for this reading order, driven by twelve values that were constructed to sum to zero. Because they sum to zero, the Brownian motion ends the year exactly where it started, and therefore the standard process in logarithms ends the year sitting exactly on its own drift line at Rs 106.18/-. The landing on the drift line is a deliberate construction rather than a coincidence, and it makes the arithmetic that follows unusually clean.

Now suppose a year of that process is recorded and the level at the horizon comes out at Rs 101.01/- rather than Rs 106.18/-. The shortfall is Rs 5.18/-. In logarithms that is a gap of 0.05. Three candidates, and only three.

All three explanations produce the identical year-end number, so the level at the horizon cannot tell them apart. The path can, instantly. The decomposition does not identify which of the three it was. The decomposition establishes that the list is three long and complete, and completeness turns an open question into a three way test run by looking at the picture.

The everyday version is a bathroom scale that reads two kilogrammes heavy. There are only so many places the error can live: the calibration is off by a constant, the mechanism is noisy, or somebody leant on it once. Knowing the list is short is what allows a check in a minute rather than a replacement of the scale. The decomposition is the same service performed for a process, with the added comfort that the list is not merely short but provably complete.

What does the definition rule out, and where does that bite?

One exclusion is where the next several years of this subject come from, and it hides in the least conspicuous clause of the four.

Line three requires that the distribution of a change depends only on the length of the stretch. Read it again with an eye for what it forbids rather than what it permits. Line three says a one month change in January must follow the same law as a one month change in September. Not a similar law. The same one. The requirement is called stationary incrementsThe distribution of a change depends only on how long the stretch was, never on when it happened, so a one month change in January follows the same law as a one month change in September., and it is the quietest of the four lines and by far the most restrictive.

Now consider a process whose variance rate is 0.04 a year for the first six months and 0.09 a year for the second six. Nothing about it is exotic. Its increments are still independent, its starting level can be set anywhere, and its path is still perfectly continuous. But a one month change in the first half carries a variance of 0.003333 and a one month change in the second half carries 0.007500, and those are two different distributions over stretches of identical length. Line three is broken. The process is not in the class, and no amount of rewriting will put it there.

In practice the exclusion bites harder still. A process whose variance rate is itself random, moving through time under its own dynamics, breaks line three for the same reason and more thoroughly. The distribution of a change now depends on the level the variance happens to have reached. This class excludes every process whose volatility moves through time. The exclusion is the reason a whole later reading order in this subject exists. That later treatment does not extend the decomposition; it leaves the decomposition behind, because the three parts stop being constants and the result they belong to stops applying.

Why is this the right level of generality for jumps?

Generality is not automatically a virtue. A description broad enough to include everything says nothing, and a description narrow enough to be sharp usually covers one case. The question worth asking about any general class is what it buys and what it costs, in that order and concretely.

The class buys the completeness result. If the class had been drawn wider, by dropping stationarity as well, the decomposition would simply be false and there would be nothing to state. If it had been drawn narrower, by keeping continuity, the class would be back to Brownian motion alone and the counting process would sit outside, along with everything else in this reading order that breaks. So the boundary is not a matter of taste. The boundary sits at the widest point where the three part statement remains true, and that is what makes it worth learning as a boundary rather than as a category.

The cost is one clause. Processes whose randomness changes character through time are not members of the class. Losing them is a real cost, and not a small one.

There is a second gain that is easy to miss. The class is closed under additionAdding two independent processes from the class produces another process in the class, with the three parts of the sum being the parts of each added together.. Add two independent processes from the class and the sum is another process in the class, whose drift is the sum of the two drifts, whose variance rate is the sum of the two variance rates and whose arrivals are the two streams of arrivals merged. Closure is what makes the jump diffusion legitimate as an object rather than a construction requiring separate justification: a process with drift and continuous randomness is added to a process with jumps, and the sum is in the class by the closure alone. The three columns of the table are additive, so a process built by adding two members has its slots filled by adding their slots.

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

Very few people prove any of this. Plenty are handed a model with an unfamiliar name and thirty printed sides of notation and have to decide, quickly, what kind of object it is and where its behaviour is coming from. The four questions below do most of that work and none of them needs the code, the data or the fitted parameters.

1. Write down the three slots before reading anything else. Ask what the drift rate is, what the variance rate of the continuous part is, and what the arrivals look like. If the notes answer all three with constants, the object is a member of the class and the decomposition applies.
   Three constants means three slots and no fourth. Anything unexplained afterwards has to be coming from one of them.
2. Check whether any of the three quantities is written with a time subscript. A variance rate carrying a time subscript is the whole exclusion in one symbol. It means the rate moves, so the increments are not stationary, so this class does not contain the object however much it looks like it should.
   One subscript is the difference between a triplet and three functions, and between a completeness result and no result at all.
3. Convert the triplet to a different frequency and see whether it survives. An annual drift of 0.06 has to be 0.005 a month and an annual variance rate of 0.04 has to give 0.003333 a month. If the notes quote monthly figures that do not divide cleanly, either the process is outside the class or somebody has converted with a rule of thumb.
   Division by twelve is the only permitted conversion here. Anything else is a signal to stop and ask.
4. When a reading surprises, name which of the three slots stands accused. This is the check that saves the most time. Saying the model behaves oddly starts an open ended search. Saying the shortfall of Rs 5.18/- is either a drift of 0.01, an ending Brownian value of minus 0.25 or one arrival at the factor 0.951229 starts a three way test that the path itself settles.
   A complaint that does not name a slot has not used the decomposition at all, and the decomposition is free.

The second check is the difference between an hour and a week, and it is the one worth internalising. The check costs one glance at the notation, and it settles whether the rest of the material being read is even relevant to the object at hand.

One last note on scope prevents a wrong inference. No jurisdiction sets the definition of a process. The four lines are the same four lines everywhere, and the completeness of the three part split is the same result in every country and every decade.

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