It is a well known fact, that it is very difficult to estimate the drift of a Brownian motion with drift from looking at a single path over a finite interval $[0, T]$. Is it the case with Levy processes with jumps?

Assume we are given a Levy process with drift $\gamma$ and Levy measure $\nu$, with no Gaussian component. And we are given a single path over a finite interval $[0, T]$ of this process. Is it possible to determine $\gamma$ without knowing the parameters?

If the process is compound Poisson then it seems that the answer is YES - because there are intervals of strictly positive length without jumps, which we can use to read the drift. What is the answer for processes with infinite activity? Finite/infinite variation?

  • $\begingroup$ When you say that you are given a path of the process, I think that you mean over a finite time horizon only, otherwise it is always possible to determine the drift, even with a Brownian component. $\endgroup$ – George Lowther Nov 23 '11 at 19:35
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    $\begingroup$ If you are given the Levy measure then, yes, you can ways determine the drift by subtracting out the compensated jumps, which converges almost surely and leaves you just with the constant drift. $\endgroup$ – George Lowther Nov 23 '11 at 19:40
  • $\begingroup$ Hi @George, I added the information that the interval is finite - this was obviously my intention. Is your comment still true, even if the process has infinite activity? $\endgroup$ – Grzenio Nov 23 '11 at 20:04
  • $\begingroup$ @Grzenio: Yes, every Levy process is the sum of (i) A Brownian motion term, (ii) the sum of jumps bigger than 1, (iii) the sum over compensated jumps less than 1 and (iv) a constant drift. The only reason you split out the jumps bigger than one is because they might not be integrable, so that they can't be compensated. The sum in (iii) converges almost-surely (with a little care about the order of summation), but not absolutely. Furthermore, the compensator of the jumps is determined by the Levy measure. $\endgroup$ – George Lowther Nov 23 '11 at 21:26
  • $\begingroup$ @Grzenio: Protter and Jacod have written a book about semi martingale discretizations with numerous LLN and CLT which might have application for Lévy processes. Though I think that there is work to do to be able to apply their results to a practical case. Best regards. Here is a (commercial) link amazon.ca/Discretization-Processes-Jean-Jacod/dp/3642241263 $\endgroup$ – The Bridge Nov 24 '11 at 10:22

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