The Levy-Khintchine Formula

Because of a string of coincidences involving my choice of courses for Part III and various lecturers’ choices about course content, I didn’t learn what a Levy process until a few weeks’ ago. Trying to get my head around the Levy-Khintchine formula took a little while, so the following is what I would have liked to have been able to find back then.

A Levy process is an adapted stochastic process started from 0 at time zero, and with stationary, independent increments. This is reminiscent, indeed a generalisation, of the definition of Brownian motion. In that case, we were able to give a concrete description of the distribution of $X_1$ . For a general Levy process, we have

$X_1=X_{1/n}+(X_{2/n}-X_{1/n})+\ldots+(X_1-X_{1-1/n}).$

So the distribution of $X_1$ is infinitely divisible, that is, can be expressed as the distribution of the sum n iid random variables for all n. Viewing this definition in terms of convolutions of distributions may be more helpful, especially as we will subsequently consider characteristic functions. If this is the first time you have seen this property, note that it is not a universal property. For example, it is not clear how to write a U[0,1] random variable as a convolution of two iid RVs. Note that exactly the same argument suffices to show that the distribution of $X_t$ is infinitely divisible.

It will be most convenient to work with the characteristic functions

$\mathbb{E}\exp(i\langle \lambda,X_t\rangle).$

By stationarity of increments, we can show that this is equal to

$\exp(-\Psi(\lambda)t)\quad\text{where}\quad \mathbb{E}\exp(i\langle \lambda,X_1\rangle)=:\exp(-\Psi(\lambda)).$

This function $\Psi(\lambda)$ is called the characteristic exponent. The argument resembles that used for Cauchy’s functional equations, by dealing first with the rationals using stationarity of increments, then lifting to the reals by the (right-)continuity of

$t\mapsto \mathbb{E}\exp(i\langle \lambda,X_t\rangle).$

As ever, $\Psi(\lambda)$ uniquely determines the distribution of $X_1$ , and so it also uniquely determines the distribution of Levy process. The only condition on $\Psi$ is that it be the characteristic function of an infinitely divisible distribution. This condition is given explicitly by the Levy-Khintchine formula.

Levy-Khintchine

$\Psi(\lambda)$ is the characteristic function of an infinitely divisible distribution iff

$\Psi(\lambda)=i\langle a,\lambda\rangle +\frac12 Q(\lambda)+\int_{\mathbb{R}^d}(1-e^{i\langle \lambda,x\rangle}+i\langle \lambda,x\rangle 1_{|x|<1})\Pi(dx).$

for $a\in\mathbb{R}^d$ , Q a quadratic form on $\mathbb{R}^d$ , and $\Pi$ a so-called Levy measure satisfying $\int (1\wedge |x|^2)\Pi(dx)<\infty$ .

This looks a bit arbitrary, so first let’s explain what each of these terms ‘means’.

$i\langle a,\lambda\rangle$ comes from a drift of $-a$ . Note that a deterministic linear function is a (not especially interesting) Levy process.
$\frac12Q(\lambda)$ comes from a Brownian part $\sqrt{Q}B_t$ .

The rest corresponds to the jump part of the process. Note that a Poisson process is an example of a Levy process, hence why we might consider thinking about jumps in the first place. The reason why there is an indicator function floating around is that we have to think about two regimes separately, namely large and small jumps. Jumps of size bounded below cannot happen too often as otherwise the process might explode off to infinity in finite time with positive probability. On the other hand, infinitesimally small jumps can happen very often (say on a dense set) so long as everything is controlled to prevent an explosion on the macroscopic scale.

There is no canonical choice for where the divide between these regimes happens, but conventionally this is taken to be at $|x|=1$ . The restriction on the Levy measure near 0 ensures that the sum of the squares all jumps up some finite time converges absolutely.

$\Pi\cdot 1_{|x|\geq 1}$ gives the intensity of a standard compound Poisson process. The jumps are well-spaced, and so it is a relatively simple calculation to see that the characteristic function is

$\int_{\mathbb{R}^d}(1-e^{i\langle \lambda,x\rangle})1_{|x|\geq 1}\Pi(dx).$

The intensity $\Pi\cdot 1_{|x|<1}$ gives infinitely many hits in finite time, so if the expectation of this measure is not 0, we explode immediately. We compensate by drifting away from this at rate

$\int_{\mathbb{R}^d}x1_{|x|<1}\Pi(dx).$

To make this more rigorous, we should really consider $1_{\epsilon<|x|<1}$ then take a limit, but this at least explains where all the terms come from. Linearity allows us to interchange integrals and inner products, to get the term

$\int_{\mathbb{R}^d}(1-e^{-i\langle \lambda,x\rangle}+i\langle\lambda,x\rangle 1_{|x|<1})\Pi(dx).$

If the process has bounded variation, then we must have Q=0, and also

$\int (1\wedge |x|)\Pi(dx)<\infty,$

that is, not too many jumps on an |x| scale. In this case, then this drift component is well-defined and linear $\lambda$ , so can be incorporated with the drift term at the beginning of the Levy-Khintchine expression. If not, then there are some $\lambda$ for which it does not exist.

There are some other things to be said about Levy processes, including

Stable Levy processes, where $\Psi(k\lambda)=k^\alpha \Psi(\lambda)$ , which induces the rescaling-invariance property: $k^{-1/\alpha}X_{kt}\stackrel{d}{=}X$ . The distribution of each $X_t$ is then also a stable distribution.
Resolvents, where instead of working with the process itself, we work with the distribution of the process at a random exponential time.

Gaussian tail bounds and a word of caution about CLT (eventuallyalmosteverywhere.wordpress.com)

8 thoughts on “The Levy-Khintchine Formula”

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Abdelkarim on January 23, 2014 at 12:17 pm said:

So, first thanks for this note. Second, I would say that’s i’m not a specialist of stochastics calculus but i’m interested to this mathematical branch. So, I have just a small question and it’s as follows: can we define a similer calcul when psi(lambda)===psi(t,lambda).

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Henk on April 5, 2015 at 8:14 pm said:

Levy-Khintchine

\Psi(\lambda) is the characteristic function -> \Psi(\lambda) is the characteristic component.

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Israel Leibovici on June 15, 2017 at 11:49 am said:

Hi Dominic!

I have seen papers where the characteristic function (phi) for certain Lévy processes X_t have been defined as E(exp^{lambda*X_t}) = exp{phi(lambda)}. For what kind of Lévy processes does this apply?

Many thanks – Issy

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cr1stal on October 15, 2019 at 7:32 am said:

so these levy processes are a 3-manifold. the semi-martingales form a proper orthogonal subgroup so you can shown that the quotient group is a vector space and the remainder is an ideal. note the zeroes of the riemann zeta function are the annihilator in this ring.

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Thomas Rae on July 26, 2020 at 5:32 pm said:

this is an excellent site. that gamma term we need to transform from the second derivative over the density to the arrival intensity integrated around the boundary and use the cauchy residue theorem. the arrival times are memoryless you need to show its positive definite symetric self adjoint with 0.5*sigma^2*t on the diag. note the real part of this is a semi-martingale and satisfies the normal flow equations with the -0.5*sigma^2*t the drift in the stochastic exponential to make it a martingale. by convolution of the gamma characteristic over the rationals you find the complex covariance matrix defines an operator the spectrum of which is the so-called zeroes of the riemann zeta function. the critical strip are those correlation terms which are in the spectrum but are not eigenvalues.

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The Levy-Khintchine Formula

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