All Questions
Tagged with levy-processes stochastic-processes
37 questions
1
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31
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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 (...
- 305
1
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0
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58
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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{...
- 305
5
votes
1
answer
202
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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 ...
- 807
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36
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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 ...
- 393
1
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1
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50
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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] \ : \...
- 619
1
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0
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140
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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 ...
- 19
3
votes
1
answer
137
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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, \...
- 133
0
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1
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450
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A complex 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-...
- 13
2
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0
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126
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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
2
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1
answer
503
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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+\...
- 123
0
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0
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68
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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. ...
1
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0
answers
175
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Interpretation of the Lévy measure of an infinitely divisible random vector
We know that a random vector $X$ is infinitely divisible (ID) if for all $n \in \mathbb N$, there exist $X_1^n,..., X_{n}^n$ i.i.d. random vectors such that:
\begin{equation}
X = X_1^n + ...+ X_n^...
- 13
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1
answer
72
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Exceedance distribution of Levy process
Consider a Levy process $L(t)$ with linear drift $-1$, no Brownian motion component, and Poisson jumps at rate 2 with size Uniform($0, 1$), and with $L(0)=0$. This process has zero mean drift.
Let $\...
- 294
1
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0
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142
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What are the Lévy processes with specific increments?
It is known that the increment of the Wiener process $W$ is drawn from a Gaussian distribution, i.e. $\Delta W \sim \mathcal{N}(0, \delta t)$.
I wonder what are the Lévy processes with increments from ...
- 11
1
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47
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How do we need to argue in this step of the Itō-Lévy-Khintchine decomposition?
Let
$E$ be a $\mathbb R$-Banach space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$-...
- 167
2
votes
0
answers
128
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Proof of the Lévy–Itō decomposition in this paper
Let
$E$ be a normed $\mathbb R$-vector space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$...
- 167
0
votes
1
answer
134
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Expectation of killed subordinator at first-passage time
I am reading Fluctuations of Levy Processes with Applications by A.E. Kyprianou and I am having struggles understanding a part in the proof of theorem 5.6. Let $Y$ be a subordinator and $\mathbf{e}$ ...
- 103
1
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0
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191
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Characterization of Poisson random measure in terms of Laplace transform
Let $(E,\mathcal E)$ be a measurable space and $\mu$ be a measure on $(E,\mathcal E)$.
A random measure $\pi$ on $(E,\mathcal E)$ is called Poisson with intensity $\mu$ if
$\pi(B)\sim\operatorname{...
- 167
0
votes
0
answers
74
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What is the sufficient and necessary condition for Blumenthal-Gettor index = 0?
This question comes from the following paper 1961(Blumenthal)
Let us consider a Levy process $X$ whose Levy triplet is $(a,s,\nu)$. According the above paper, Blumenthal-Gettor index is given by $$\...
2
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96
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Invariant measures of Levy S.D.Es
Suppose we call a real valued stochastic process $\{Z_t\}$ to be distributed as ${\cal S}\alpha{\cal S}(\sigma)$ if each of the characteristic functions is $\phi_{Z_t}(u) = \exp\left\{-t\vert \sigma u ...
- 2,246
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2
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230
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Characterization of the generator of a Lévy process using martingale problems
Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...
- 167
1
vote
1
answer
168
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Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$
Let $E$ be a $\mathbb R$-Banach space, $\mathcal M_1(E)$ (resp. $\mathcal M_1^\infty(E)$) denote the set of probability measures (resp. infinitely divisible probability measures) on $E$, $\varphi_\mu$ ...
- 167
0
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0
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150
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Define the convolution root of probability measures on a measurable group
Let $(G,\mathcal G)$ be a measurable group and $\nu^{\ast k}$ denote the $k$th convolution power of a probability measure $\nu$ on $(G,\mathcal G)$ for $k\in\mathbb N$.
Remember that a probability ...
- 167
0
votes
2
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368
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If $\mu$ is an infinitely divisible probability measure on $[0,\infty)$, then the Lévy measure of $\mu$ is the vague limit of $n\mu^{*1/n}$
If $\nu$ is a finite measure on $(\mathbb R,\mathcal B(\mathbb R))$, let $\nu^{\ast k}$ denote the $k$-fold convolution¹ of $\nu$ with itself for $k\in\mathbb N_0$, $$\exp(\nu)\mathrel{:=}\sum_{k=0}^\...
- 167
-1
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1
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92
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Is the distribution of a Banach space valued Lévy process uniquely determined by its characteristic function?
Let $E$ be a $\mathbb R$-Banach space. Remember that if $\mu$ is a finite measure on $\mathcal B(E)$ then $$\Phi_\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi(x)}$$ is ...
- 167
3
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0
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136
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An integral involving Levy process with no positive jumps
Let $L_t$ be a Levy process that has no positive jumps, but is not strictly decreasing, i.e
$$
L_t = \gamma t + \sigma B_t + J_t,
$$
where $B_t$ is a Brownian motion, $J_t$ is a pure jump process with ...
- 103
0
votes
0
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62
views
Hitting order of sets by a Lévy process
Let $X$ be a transient Lévy process on $\mathbb R$, and $B\subseteq \mathbb R$ a Borel set with first hitting time $T_B = \inf \left\{t>0 : X_t\in B\right\}$. For Borel $A\subseteq B$, can anything ...
- 39
2
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1
answer
110
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Probability of exiting on the boundary for a monotone Lévy-type process
Let the continuous function $\ell:\mathbb R \times(0,\infty)\to[0,\infty)$ be a Lévy-type kernel, such that
$$
\sup_{x}\int_0^\infty \min\{1,y\}\ell( x, y)\,dy<\infty,
$$
and suppose that $\...
- 213
4
votes
1
answer
113
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escape points of Levy processes
Suppose $D$ is a domain in $\mathbb{R}^d$, $x\in D$. $X_t$ is a Levy process with Lévy triplet ${\displaystyle (b,0,\mu )}$ . Can one give a brief proof for:
$$
\mathbb{P}_x(X_{\tau_D^-}\in \partial ...
- 515
2
votes
2
answers
876
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Monotone convergence theorem for stochastic integrals
I was wondering if there exists an equivalent of a monotone convergence theorem for stochastic integrals. I looked into plenty of books and papers, but I haven't found anything useful. I would expect ...
- 121
3
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0
answers
101
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A dependent and discrete version of the Komlós-Major-Tusnády theorem
The well-known Komlós-Major-Tusnády approximation gives sharp speed of convergence of a uniform empirical process to a Brownian bridge. Here I am considering how to approach a similar problem with ...
- 773
2
votes
0
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146
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Modulus of continuity of Lévy process as jump size tends to zero
While reading Kallenberg's "Foundations of Modern Probability Theory", 2nd edition, the following question regarding an argument in the proof of Lemma 15.19 occurred to me.
Let $X_n(t)$ be a sequence ...
- 87
2
votes
1
answer
354
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Itô Formula for Hilbert space-valued Lévy processes
I know there are Itô formulas for cylindrical Brownian motions with values in a Hilbert space and Itô formulas for Lévy processes in $\mathbb{R}^d$. My question is:
does there exist an Itô formula ...
- 5,405
2
votes
1
answer
418
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Version of Donsker-Invariance-Principle
Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$...
- 257
3
votes
2
answers
3k
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Lévy measure and Lévy process
A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying
$$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$
A Lévy process can be characterized by triples $(b, A, \...
- 1,399
2
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0
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278
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Radon-Nikodym for continuous time processes
Likelihood theory for statistical inference concerning stochastic processes in continuous time are well used. How ever i've found no real literature concerning the fundamentals.
What is know from ...
- 257
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1
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215
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Definition: Grigelionis Process? [closed]
Background
I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and ...
- 5,405