Lévy processes, which are defined as having stationary and independent increments, were introduced in the previous post. It was seen that the distribution of a d-dimensional Lévy process X is determined by the characteristics via the Lévy-Khintchine formula,
The positive semidefinite matrix describes the Brownian motion component of X, b is a drift term, and is a measure on such that is the rate at which jumps of X occur. Then, equation (1) gives us the characteristic function of the increments of the process.
In the current post, I will investigate some of the properties of such processes, and how they are related to the characteristics. In particular, we will be concerned with pathwise properties of X. It is known that Brownian motion and Cauchy processes have infinite variation in every nonempty time interval, whereas other Lévy processes — such as the Poisson process — are piecewise constant, only jumping at a discrete set of times. There are also purely discontinuous Lévy processes which have infinitely many discontinuities, yet are of finite variation, on every interval (e.g., the gamma process).
Let’s start with the simple case of processes with finitely many jumps in each bounded time interval. The proof is given below in the more general case of non-stationary independent increments (see Lemma 6).
- With positive probability, X has finitely many jumps in some time interval (s,t] with .
- Almost surely, X has finitely many jumps in every bounded interval.
Furthermore, if these conditions hold then the number of jumps in the time interval (s,t] has the Poisson distribution with parameter .
For example, this includes homogeneous Poisson processes of rate , where has the Poisson distribution of rate . In that case, the Lévy measure is just , where represents the Dirac measure at 1. More generally, we can construct pure-jump Lévy processes as follows. Fix a constant and probability measure on with . Then, let be a sequence of exponentially distributed random variables with parameter and be -valued variables with measure , all of which are independent. Setting , then is a homogeneous Poisson process of rate . We can define the piecewise-constant process
which can be seen has stationary independent increments, so it a Lévy process. Then, X is known as a compound Poisson process of rate and jump distribution . Its characteristic function can be computed in terms of the characteristic function of .
Substituting in the expression for as an integral with respect to gives,
Comparing this with the Lévy-Khintchine formula shows that X has Lévy characteristics , where and
Conversely, consider any Lévy process X with characteristics , where is finite. By the decomposition into continuous and purely discontinuous parts, we can write this as
where W is a d-dimensional Brownian motion with covariance matrix , is given by (2) and Y is a Lévy process with characteristics . Letting and be the probability measure , we see that Y is a compound Poisson process of rate and jump distribution . This, then, describes all Lévy processes whose jumps occur at a finite rate.
We can also give conditions for a Lévy process to have finite variation on bounded intervals. The proof is left until Lemma 7 below.
- With positive probability, X has finite variation over some time interval (s,t] with .
- Almost surely, X has finite variation on every bounded interval
Furthermore, in this case, we have
where is given by (2).
Property (3) is particularly interesting in the case where is infinite. In that case, X jumps infinitely often but has finite variation in every nonzero bounded time interval. When X is an FV process, it is often convenient to use the decomposition (4), so that we break it up into a constant drift term and a pure jump process Y satisfying . Such pure jump processes have a particularly simple expression for the characteristic function ,
This follows from Theorem 2 by noting that, in this case, so that the Lévy-Khintchine formula simplifies. Comparing with the form of the Lévy-Khintchine formula given in (1), we see that this avoids the rather messy term in the integral. However, some purely discontinuous Lévy processes, such as the Cauchy process, do have infinite variation on bounded time intervals, so the more general form in (1) is used.
Real valued and nondecreasing Lévy processes starting from 0 are known as subordinators. In particular, these are finite variation processes, so (3) is satisfied. Subordinators are easily described in terms of their characteristics.
Then, X is nondecreasing if and only if is supported by the positive reals and is nonnegative.
Proof: If X is nondecreasing then its jumps must be nonnegative, so . Also, (4) shows that X has continuous part which must be nondecreasing, so . Conversely, if and then X has nonnegative jumps and (4) shows that X is nondecreasing.
An example of a subordinator can be constructed from hitting times of a standard Brownian motion B started from 0. For each , let be the stopping time
Considering a as a time index, the process is right-continuous and increasing. By the strong-Markov property, is also a standard Brownian motion independently of which passes through level b at time , so . From this, we see that is a subordinator.
At each time , the Brownian motion B is almost surely strictly less than its maximum , which implies that t is in the union of the intervals . So, the union almost surely covers almost all of the interval . This means that is a pure jump process. We can compute its Lévy measure. We have and, by the reflection principle, this has probability exactly twice that of being greater than a. So, in the limit as a goes to zero,
The Lévy measure of satisfies , so . Also, using the fact that is a Brownian motion hitting a at time , we see that . In particular, the sum of two independent copies of has the same distribution as . This is equivalent to saying that is a stable random variable with stability parameter 1/2. Lévy processes whose increments have stable distributions are known as stable processes and, in particular, is a stable subordinator. Writing the moment generating function as , the Lévy-Khintchine formula can be used to calculate , from which we obtain . This verifies the well-known formula for Brownian motion hitting times,
Other examples of subordinators include the gamma process. A gamma process X with mean and variance per unit time is a Lévy process with such that has the gamma distribution with mean and variance . Setting and , this has probability density function
From this, we see that it has Lévy measure . Furthermore, computing
we see that the drift term in (4) is zero, so that this is a pure jump process with . As it has the gamma distribution, the characteristic function of X is
Subordinators are often used to apply stochastic time changes to a process. If Z is a Lévy process and, independently, is a subordinator, then is another Lévy process. For example, the variance gamma process is a Brownian motion time-changed by a gamma process. Let be a standard Brownian motion with drift and be a gamma process with mean 1 and variance per unit time. As is a pure jump process, then the variance gamma process is also a pure jump process satisfying . Its characteristic function can be calculated from the characteristic functions of the normal and gamma distributions,
where . Factoring the quadratic into linear terms gives
However, this is just the product of the characteristic function of a gamma process and minus a gamma process, from which we see that variance gamma processes are the difference of independent gamma processes. Therefore, X has Lévy measure
An example of a purely discontinuous Lévy process with infinite variation over all nonzero time intervals is given by the Cauchy process, which is a stable Lévy process with stability parameter . If X is a Cauchy process, then has the symmetric Cauchy distribution at time t. The probability density function is
So, X has Lévy measure . In particular, as , inequality (3) does not hold and, consequently, X is not a finite variation process.
It is possible to determine the integrability of a Lévy process from its Lévy measure. In fact, integrability is equivalent to the apparently much weaker property of local integrability, and is also equivalent to the seemingly much stronger condition of integrability of its maximum process. The proof of the following theorem will be given in Lemma 8 below for the more general situation of inhomogeneous independent increments processes.
- X is locally -integrable.
- X is -integrable.
- is -integrable.
- The Lévy measure satisfies .
So each of the examples of Lévy processes mentioned above are integrable, with the exceptions of the 1/2-stable process of Brownian hitting times and the Cauchy process.
Note that if X is an integrable Lévy process then has independent increments of mean zero, so is a martingale. It is therefore possible, and often convenient, to decompose such processes as the sum of a constant drift term and a martingale Lévy process.
where , W is a Brownian motion with covariance matrix and M is an -integrable martingale with characteristic function where,
As , the final statement of Theorem 4 implies that the integral in (6) is well-defined. The proof of this Lemma follows quickly from previous results of these notes. First, for any , decomposition (5) follows from the decomposition of X into its purely continuous and discontinuous parts, W and M respectively. Then, M is an -integrable Lévy process with characteristics . Setting
then (6) follows from the Lévy-Khintchine formula. It only remains to show that M is a martingale and, by the independent increments property, it is enough to show that it has zero mean. Using dominated convergence to commute the differentiation with the integral in (6), we can calculate
So, differentiating (6) at a=0 gives .
Inhomogeneous independent increments processes
All of the results given above apply equally to time-inhomogeneous independent increments processes, and I will now go through their proofs at this level of generality. Throughout this section, it is assumed that X is a cadlag d-dimensional process with independent increments, and which is continuous in probability. As previously shown, the increments of such a process have characteristic function of the form
Here, is a map from time t to a symmetric matrix with and is increasing in the sense that is positive semidefinite for all . Also, is a continuous function starting from zero, and is a Borel measure on such that for all , and
In the case where X also has stationary increments, so that it is a Lévy process, then are related to the Lévy characteristics by , and . In that case, equation (7) reduces to the standard Lévy-Khintchine formula.
We start by giving a proof of Theorem 1 which, just being a re-statement of results previously covered in these note, is particularly simple.
- With positive probability, X has finitely many jumps.
- With probability one, X has finitely many jumps.
- is finite.
In this case, the number of jumps of X is Poisson distributed with rate .
Proof: Letting be the total number of jumps of , then this is just a restatement of the fact that is Poisson distributed with rate whenever is finite, and is almost surely infinite whenever is infinite.
Theorem 1 is an immediate consequence of this. If X is a Lévy process with characteristics , then the first statement of Theorem 1 implies that there is a non-trivial time interval [s,t] on which, with positive probability, X has finitely many jumps. Applying Lemma 6 to X restricted to this interval implies that is finite. So, and it follows that is finite for all and, again applying Lemma 6, the number of jumps in any finite time interval [s,t] is almost surely finite with the Poisson distribution of rate .
I now give a proof of the following generalization of Theorem 2.
- With positive probability, V is finite.
- With probability one, V is finite.
In this case, setting , then
for all real a.
Proof: This result actually holds in the generality of Poisson point processes although, here, we are only concerned with the application to independent increments processes. Let us start by considering the case where is finite. In that case, the identity
shows that V is almost-surely finite. So, is a well-defined independent increments process. Therefore, its characteristic function is of the form where
is a square integrable martingale. Taking expectations and letting t increase to infinity, so that , gives
as required. Note that the inequality implies that is -integrable, so this expression is always well defined.
It only remains to show that all this holds whenever V is finite with nonzero probability. Consider, for the moment, the case where is nonnegative and . Then, is uniformly bounded for all complex a with nonnegative imaginary part and, by analytic continuation (8) extends to all such a. So, we can replace a by ia to get
for all . However, this identity extends to all nonnegative f. Simply apply it to a sequence of nonnegative functions with and use dominated convergence on the left and monotone convergence on the right to take the limit as n goes to infinity. Suppose that the third property of the statement of the Lemma did not hold, so that . Then, the inequality gives for all positive a. So, the right hand side of (9) is zero. Taking the limit as a decreases to 0,
Conversely, if has nonzero probability of being finite, then we see that must be finite. Applying this to for arbitrary (not necessarily positive) f shows that the first condition of the Lemma implies the third.
To show that this result implies Theorem 2, consider a Lévy process X with characteristics . As the continuous part of its quadratic variation, is constant over any interval on which X has finite variation, the first condition of Theorem 2 implies that is zero. If X has finite variation with positive probability over an interval [s,t], Lemma 7 applied to shows that
is finite. So, . Then, applying the lemma to any interval [0,t] shows that is almost surely finite, so is well defined. As this is a continuous Lévy process with zero quadratic variation, it is simply of the form for a constant . So, X almost surely has finite variation over all bounded time intervals. To calculate c, we can apply (8) with to obtain
Finally, we give a proof of Theorem 4 for independent increments processes.
Proof: First, consider any nonnegative measurable function such that
is finite for all . Then, taking , we know that is a martingale so, by optional stopping,
for all stopping times . Note that this extends to all nonnegative measurable . Just apply it to a sequence increasing to such that is finite, and use monotone convergence as n goes to infinity. In particular, consider . In that case . If property 1 holds so that X is locally -integrable, then Y is locally integrable. Therefore, there is a stopping time with and . Applying (11) to this gives
So is finite and (10) holds.
Conversely, supposing that (10) holds, it only remains to be shown that is -integrable for each fixed time t. To do this, fix a constant and define stopping times and
for . For each time , there almost surely exists an n with , so . Therefore, we can bound in the norm by
Noting that is bounded by whenever and zero otherwise, its norm satisfies the bound
The terms on the right hand side can be bounded as follows. By the independent increments property, for all times ,
Using to denote the term on the right hand side, we can assume that K has been chosen large enough that this is strictly less than 1 (actually, this is true for all positive K). Then, as the space-time process is a Feller process and, hence, satisfies the strong Markov property, this inequality holds when s is replaced by a stopping time. Replacing s by gives . So, and, by induction, we see that is bounded by .
Again using the independent increments property for ,
Denote the right hand side by c which, by (10), is finite. Applying the strong Markov property again to replace s by , this gives
Putting these bounds back into (13),
Finally putting this back into (12) gives the required bound