Questions tagged [levy-processes]
Theory and applications of Lévy processes (stochastic processes with stationary and independent increments): e.g. path properties, stochastic differential equations driven by jump-type processes, fluctuation theory of Lévy processes, queuing theory.
69 questions
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Question regarding this exercise on Cramér-Lundberg model with added Brownian motion
I've been trying to do this little exercise for the past couple of days but I really can't proceed at all. It regards calculating this: Given the classic Cramér-Lundberg processes with added Brownian ...
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Tail estimates on $L^\infty([0,T];X)$-norm of Banach space valued S$\alpha$S Lévy process
As described in the title, I am looking for a reference for tail estimates on suprema of Lévy processes. In one dimension, it is well known that for a real-valued symmetric $\alpha$-stable Lévy ...
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$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
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Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)
Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation:
$$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
- 321
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Independent stationary increment process but with finite propagation speed
Intuitively, standard Brownian motion has infinite propagation speed, as it has a non-zero probability of reaching any point in any arbitrarily short time. This is due to the fact that the probability ...
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Interpretation of Lévy process with signed Lévy measures
Suppose that I have a non-decreasing, pure jump Lévy process of finite variation $X$ with Lévy measure $\pi$. The Lévy measure is then supported on $(0,+\infty)$. Suppose that the Lévy measure is a ...
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Injectivity of a convolution operator
Let $p,\mu,\nu$ be probability density functions on
$\mathbb{R}$ such that
$$
\int_{\mathbb{R}}p(y-x) \nu(y) \, dy=\mu(x).
$$ Now, consider the operator $T:L^2(\mu)\to L^2(\nu)$ such that $$ Tf=f*p.$$ ...
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translation invariance of expectation value of hit counting variable for Lévy process
Let $(X_t)_{t \in [0, \infty)}$ a $\mathbb{R}$- valued
Markov process (in my question I'm primary interested in dealing with Lévy process), $s, a, u >0$,
$I(a) :=
\{[k \cdot a, (k+1) \cdot a] \ : \...
- 705
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Ask assistance for finding K. Sato - Lévy Processes on the Euclidean Spaces
The paper me and my professor want is called K. Sato (1995) Lévy Processes on the Euclidean Spaces, Lecture Notes, Institute of Mathematics, University of Zurich.
I tried to find the paper on the ...
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Thinning of (mixed) binomial point process
Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...
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A question related to a certain convergence of Lévy measures
Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
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Supremum process of a Cauchy RV
I've asked the same question on stats.stackexchange a week ago to no avail, so here we go again:
Suppose $X_i$ are $\mathrm{Cauchy}(0,~\gamma)$ IID RV's. Does an expression exist for the CDF of the ...
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A question related to the jumps of a Levy process
The Lévy–Khintchine formula says that any Lévy process, $X=(X(t), t \geq 0)$, has a specific form for its characteristic function. More precisely, for all $t \geq 0$, $u \in \mathbb R^d$:
$$
\mathbb{E}...
- 121
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Stationary Distribution of Langevin Dynamics driven by Lévy Process
Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\...
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Step in the derivation of the total idle time distribution of an M/G/1 queue
I'm trying to work my way through the proof of Thm. 1.11 in Kyprianou's Introductory Lectures on Fluctuations of Levy Processes with Applications but really struggle to understand the following step. ...
