Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A Levy subordinator is an finite variation Levy process with non-negative drift and positive jumps. The Levy exponent is given by

$$\phi(\lambda) = \gamma \lambda + \int_0^\infty ( 1 - e^{-\lambda s} ) \nu(ds)$$

where $\gamma>0$ is the drift of the subordinator and $\nu$ is the jump measure (Levy measure). If the jumps are a compound Poisson process with (net) jump intensity $\alpha$ and jump-size distribution $\mu$ then $\nu = \alpha \mu $ and the levy exponent becomes

$$\phi(\lambda) = \gamma \lambda + \alpha (1 - \widehat{\mu}( \lambda ) )$$

where $\widehat{\mu}( \lambda ) = \int_0^\infty e^{- \lambda s} \mu(ds)$.

My Questions are as follows:

  1. Given a function $\phi(\lambda)$, how do I know if there is a $\nu$ and a $\gamma$ that generates it?

  2. For a given $\phi$ is the pair that generates it $(\gamma,\nu)$ unique?

  3. Assume the jumps are a compound Poisson process. If you are given $\phi(\lambda)$ can you find $\alpha$ and $\gamma$? Finding $\alpha$ and $\gamma$ would uniquely determine $\widehat{\mu}( \lambda )$ and allow us to reconstruct $\mu(ds)$ from the inverse Laplace transform. Then $\nu(ds) = \alpha \mu(ds)$.

  4. More generally, given $\phi(\lambda)$, can you find $\nu$ and $\gamma$.

The reason for these questions is that I am going to numerically construct $\phi(\lambda)$ from data. Ideally, I would like to then construct $\gamma$ and $\nu$ (or $\alpha$ for a Poisson process) as well. At this point, it isn't clear to me that I actually need $\gamma$ and $\nu$ for my calculations. It may be that $\phi(\lambda)$ is enough (this project is in its nascent stage at the moment). But, even if I don't need $\nu$ and $\gamma$ I am curious to see if I can construct them. And an existance and uniqueness result would definitely strengthen my paper.


So, I have a partial answer to the construction of $(\gamma,\nu)$ from $\phi$. Clearly $\gamma = \lim\limits_{\lambda \to \infty} \phi(\lambda)/\lambda$.

Still looking for a construction of $\nu$ at the moment.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Once you get $\gamma$, you can calculate

$\frac{\phi'(\lambda)-\gamma}{\lambda}=\int_0^\infty e^{-\lambda s}\nu(ds)$

and $\nu$ can be obtained by Laplace inversion.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.