Jump-Interval Distributions in Lévy Processes
Summary
The document explains how the Lévy measure determines whether successive jump times and their gaps can be treated as discrete events. When the Lévy measure has finite total mass, jumps form a compound Poisson process, with rate equal to that total mass; the waiting times between jumps are then independent exponential variables. The expected gap and its variance are finite when this rate is positive and finite.
When the Lévy measure has infinite total mass, infinitely many small jumps can occur in every time interval, so there is no ordinary sequence of all jump times with positive gaps. In that case, the proposed inter-jump interval is not meaningful without restricting attention to jumps above a size threshold. The answer points to standard Lévy-process references but does not derive the general theory or analyze the Gamma-process example in detail.
Key ideas
- A finite Lévy measure gives a compound Poisson jump process with rate equal to its total mass.
- Successive jump gaps in that finite-activity case are independent exponential random variables.
- An infinite Lévy measure implies infinitely many small jumps in every interval, so ordinary gaps between all jumps are not defined.
- Restricting jumps by a size threshold can make jump times meaningful even when the full process has infinite activity.
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# The distribution of jump gaps for Levy processes
# The distribution of jump gaps for Levy processes
Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps happen.
There are two questions for me. First, is $\tau_{i}$ well defined?. Second, the answer of first one is yes, we know $\tau_{i}$ are i.i.d. Then how to find the distribution of $\tau_{i}$? Can we prove the expectation of $\tau_{i}$ is finite or even its variance is finite according to the information of Levy measure $\nu$?
I can do this when the Levy process $X_{t}$ is Poisson process or negative binomial process. I have difficulty when the probability density function of $\nu$ is continuous. For example, the case that $X_{t}$ is a Gamma process.
Any references would be very appreciated.
## Answer by quasi (score 3)
https://quant.stackexchange.com/a/10268
This is a good shorter reference: http://www.impan.pl/CZM/tankov.pdf. Cont and Tankov have also written a longer book about modelling with Levy processes that I think is really good.
There's going to be a strong connection between the sequence of jump times and the Levy measure $\nu$. In a single unit of time, $ \nu(dx)$ is a measure (not necessarily a probability) that records the expected number of jumps of size dx.
Now, if $\nu$ is not a finite measure, then I don't think it makes sense to talk about the jump times. They're basically happening all the time. So, for example, if you tried to define $\tau_1 = \inf \{ t \geq 0 : \Delta t \neq 0 \}$, $\tau_1$ would be equal to zero. This is why I say the notion isn't meaningful.
On the other hand, suppose that $\nu$ is a finite measure. This is closer to the setting of a Poisson process, where $\nu$ is $\lambda$ times a point mass at 1, or for a compound Poisson process, $\lambda$ times the pdf of the jump size distribution. So analogously, let $\lambda' = \nu(\mathbb{R})$. This is the expected number of jumps per unit of time. Then the jump times (ignoring size) will come like a Poisson process with intensity $\lambda'$, so the individual gaps have the corresponding exponential distribution.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.