Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.
Questions tagged [levy-processes]
369 questions
2
votes
2 answers
Small time behavior of Lévy processes
Let $(X_t)_{t \geq 0}$ a Lévy process and $\varepsilon>0$. Is there anything known about the asymptotics of the probability
$$\mathbb{P}(|X_t| > \varepsilon)$$
as $t \to 0$? Obviously, by the stochastic continuity, this probability converges to $0$…
saz
- 120,083
2
votes
0 answers
Calculation of Lévy process expected value
After my last post went horribly wrong I want to do things right this time.
I'm researching Lévy process in my free time as part of my interest in stochastic processes.
Unlike different processes I've seen that the moments of Lévy process are not…
bb_1905
- 21
1
vote
2 answers
Show $X_t = X_{t-}$ a.s. for a Levy process $(X_t)_{t \geq 0}$
I started reading Sato's book on Lévy processes. On page 6 it says that for $(X_t)_{t\ge 0}$ Lévy process
$$X_t=X_{t-},$$
for any fixed $t>0$ almost surely. It is mentioned it follows from the fact that almost sure convergence implies convergence in…
Analyst77
- 61
1
vote
0 answers
Lévy–Khintchine representation for distributions
I have read that the Lévy–Khintchine representation exists for any infinitely divisible distribution. However, all the references I could find on Lévy–Khintchine representations are for Lévy processes.
But how to derive the Lévy–Khintchine…
p-value
- 474
1
vote
1 answer
How to find Lévy Triplets
I have recently started to learn about Lévy processes, and have learnt about the Lévy-Khintchine theorem. My question is about the Poisson process, it stated to have Lévy triplet (0,0,$\lambda\delta(1)$), where $\delta(1)$ is the dirac measure with…
df12
- 39
- 5
0
votes
0 answers
A Levy process which is not a compound Poisson drifting to $-\infty$
I have read a few books about L'evy processes and tried to find a concrete example such that $\{X_t\}$ is a L'evy process (but not a compound Poisson) drifting to $-\infty$ and $0$ is not regular for $(0,\infty)$. So far, all I could find are a lot…
user377704
- 421
0
votes
0 answers
Intuition on Lévy process when it had positive jumps only, no Brownian motion part and $\int_{x\in (0,1]} x \nu(dx) = \infty$.
I read Sato's section 21 in 'Lévy processes and infinitly divisible distributions (1999)' and I don't understand this one passage which is a consequence of Theorem 21.5.
It reads:
'A consequence of Theorem $21.5$ should be contemplated. A Lévy…
