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Questions tagged [levy-processes]

Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.

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Do jumps in Levy processes need to be independent of the process itself?

I have a very basic question about Levy processes. Is the process of the form $$ X_t=\sigma (X_t) B_t + \sum_{n=1}^{N_t}\eta(X_t) $$ a Levy process? Here $B_t$ is a standard Brownian motion and $N_t$ ...
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1answer
25 views

Convergence of the Laplace Exponent of a Compound Poisson Process, Lèvy Fluctuation Theory

The question is about spectrally positive Lévy processes. For certain $d, \sigma^{2} \geq 0$ and measure $\Pi_{\varphi}(\cdot)$ such that $\int_{(0,\infty)} \min \{1, x^2 \} \Pi_{\varphi}(\cdot) (dx) ...
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1answer
32 views

Does partial averaging allow moving increments in and out of an expectation?

Given a Levy process $X$ at different points in time $s$ and $t$, and if I have an expression like this: $$\mathbb{E}[X_t \cdot \mathbb{E}[X_s]]$$ I want to know if I can use partial averaging to ...
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561 views

Proving properties for the Poisson-process.

Define a Poisson process as a Levy process where the increments have a Poisson distribution with parameter $\lambda$*"length of increment". I want to prove these properties: It has almost surely ...
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75 views

How to calculate expected value of integral?

How to calculate $E \big[(\int ^{t} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^A (\int ^{t+h} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^B\big]$, where \begin{align} \tilde{L}_\alpha (t) = \...
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31 views

Inverting a Laplace transform (for a Lévy process)

Let $\psi(\theta) = c\theta + \frac{\sigma^{2}}{2}\theta^{2} - \frac{\lambda\theta}{\alpha + \theta}.$ For those who are wondering where this function comes from, $\psi$ is the Laplace exponent for a ...
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Simulating a Cauchy process

My question is how to I simulate sample paths from a Cauchy process? I know this can be done using two Brownian motions, but I am trying to do it from the basics. It's known that if we have a Levy ...
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1answer
43 views

Lévy process + scaling property $\implies$ Brownian motion

How can I show that if $\xi_t$ is a Lévy process distributed as $\xi_{t+s}- \xi_s$ for all $t,s \in [0,\infty)$ and has independence of increments, and also is distributed as $\lambda\xi_{\lambda^{-2}...
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1answer
37 views

Stochastic integral with Poisson random measure

The following is what I read in paper and I am confused by some parts. We consider a one-dimensional Itô semimartingale $X$ which is defined on some probability space $(\Omega,\mathcal F,\{\mathcal ...
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1answer
209 views

Finiteness of the number of big jumps of a Lévy process on a finite interval

Let $Y(t)$, $0 \leq t \leq 1$, be a Lévy process. Denote by $\{ \Delta Y_i \}$ the jumps of the process. I would like to show that the set $\{i: |\Delta Y_i| > r\}$ is finite almost surely. However,...
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64 views

The Levy measure of a multivariate alpha-stable Levy motion

I am having difficulties to understand the form of the Levy measure of the multivariate Levy-stable motion. Let me start by defining the one dimensional motion in order to clarify my question. The ...
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15 views

generate random numbers of Lévy distribution

I am going to generate a random step length which is drawn from a Lévy distribution $$ \textrm{Lévy } \sim u= t^{-\lambda},\;\; 1<λ ≤3 $$ How to (in computer) generate $u$ of such probability? ...
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57 views

Levy construction of Brownian motion by Haar function and Schauder function

For every $t \in [0,1]$, we set $h_0(t) = 1$, and then, for every integer $n \geq 0$ and every $k \in \{0,1,2,...,2^n-1\},$ $$h^n_k(t) = 2^{n/2}\mathbb{1}_{[(2k)2^{-n-1},(2k+1)2^{-n-1})}(t) - 2^{n/2}\...
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92 views

How to compute the levy path integral with zero potential?

In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as $$K(x_b,t_b|x_a,t_a)=\int_{x_{t_a}=x_a,...
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52 views

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 $\...
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Construction of semimartingales

Let $(B,C,\nu)$ be the characteristics of a semimartingale $\{X_t\}_t$ on $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$. If \begin{equation} B_t(\omega)=bt, \ C_t(\omega)=ct, \ \nu(\omega,dt,dx)=...
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1answer
266 views

Finding the Levy measure

I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am ...
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974 views

Jumps of Lévy process

Let $X:=(X_t)_{t\geq0}$ be a Lévy process with triple $(b,A,\nu)$. Is there any known relation between the "distribution" of its jumps and the Lévy measure $\nu$? E.g. can we express something like $\...
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1answer
64 views

Characteristic function of a Levy process

When I was reading Protter's textbook "Stochastic Integration and Differential Equations", it is stated without proof that (page 20) If we take the Fourier transform of of each $X_t$ (a levy process)....
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57 views

Book on stochastic differential equations

I'm applying for a job at a sports forecaster on the mathematical modeling side. My interview ended with the handing out of a test for which I have a week. With only a MSc thesis on Sasaki-Einstein ...
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2answers
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Levy measure of borel sets away from $0$

The following is of Philip Protter at page 26 of the book Stochastic integration and Differential equations that I have not been able to proved yet. Let $X$ a Levy process, and $\Lambda$ a borel set ...
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1answer
59 views

Characteristic function of maximum of Levy process

Nothing to add to the title, I'm looking for the characteristic function of the maximum of a Levy process, can someone help me out? Thanks
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Wiener-Hopf factorization of the characteristic function of a Levy process

Given $X_t$ a Levy process and $\Delta$ an interval of time, I have to compute the Wiener-Hopf factorization $\Phi_+$$\Phi_-$ of $$\Phi(u,q)=1-q\mathbf{E}[e^{iuX_{\Delta}}]=1-q\varphi(u)=\Phi_+(u,q)\...
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1answer
55 views

Definition of Lévy process

I know that Lévy process $\{X_t\}_{t\geq 0}$ is a stochastic process that satisfies few conditions: $\mathbb{P}(X_0 = 0) = 1.$ $X_t$ has stationary increments and $X_t$ has independent increments. ...
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98 views

Proof that the Expected Value of a Levy Distribution diverges?

If this question has already been answered, please link me because I could not find anything online. I have been using exponential Brownian motion in my models of stochastic population dynamics. The ...
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1answer
56 views

Scaling property for $\alpha-$stable subordinators

I'm trying to prove that, for a $\alpha-$stable subordinator, with $\alpha\in(0,1)$ the next equality holds:$X_{t}\overset{d}{=}t^{1/\alpha}X_{1}.$ The definition of $\alpha-$stable subordinator is ...
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1answer
41 views

Lévy's triplet of $Y_{t}=aB_{t}+Z_{t}+rt$

I'm trying to find the triplet of the next Lévy process: $$Y_{t}=aB_{t}+Z_{t}+rt,$$ where $\{B_{t}\}_{t\geq 0}$ is a standard Brownian Motion, $\{Z_{t}\}_{t\geq 0}$ is a Compound Poisson Process and $...
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149 views

Laplace exponent of a standard $\alpha-$stable subordinator

I'm trying to calculate the Laplace exponent of a standar $\alpha-$stable subordinator. An $\alpha-$stable subordinator has Lévy measure $\frac{c}{x^{1+\alpha}}dx,$ where $\alpha\in(0,1)$ and $c$ is ...
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1answer
46 views

Does the Levy process stay in any open ball at any fixed time with positive probability?

Question: Let $X$ be a $d$-dimensional Levy process. Then for every $t>0$ and $a>0$, \begin{equation}\tag{1} \mathbf P\{|X_t|<a\}>0\ ? \end{equation} The question comes from the proof ...
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1answer
58 views

Asymptotic behavior of Lévy processes

I'm trying to prove the next proposition: Let $\{X_{t}\}_{t\geq 0}$ be a non-zero Lévy process on $\mathbb{R}.$ Then it satisfies one of the following three conditions: $$i)\space\displaystyle\lim_{t\...
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1answer
37 views

Recurrent Lévy process implies $\limsup_{t\rightarrow\infty}X_{t}=\infty$ and $\liminf_{t\rightarrow\infty}X_{t}=-\infty$

Let $\{X_{t}\}_{t\geq 0}$ be a non-zero recurrent Lévy process on $\mathbb{R}.$ Then $$\space\displaystyle\limsup_{t\rightarrow\infty}X_{t}=\infty\space\text{a.s.}\space\text{and}\space\displaystyle\...
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Measurability of inferior limit and limit of a Lévy Process

I'm reading about Lévy Processes. As a definition of a recurrence and transient Lévy process we have: Def. For a Lévy process $\{X_{t}\}_{t\geq 0}$ in $\mathbb{R}^{d}$ is called recurrent if $\...
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Doubts in equivalence between recurrence and Potential measure for Lévy processes

I'm reading about Potential measures of Lévy processes. There is a theorem which proof is not totally clear to me. The teorem is the next: Suposse that $H=\mathbb{R}^{d},$ where $H$ is the spanned of ...
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41 views

Support of Potential Measure for Lévy Processes

I'm reading about Support of Potential measures for Lévy Processes. $\Sigma$ denotes the support of $U(0,\cdot),$ where $U(x,B)=\int_{0}^{\infty}P_{x}(X_{t}\in B)\,\mathrm dt$ is the potential measure ...
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1answer
32 views

Resolvent operators describe distribution of Lévy Process evaluated at independent exponential times

I'm reading about resolvent operators of Lévy processes. The definition is the next: $$U^{q}f(x)=\int_{0}^{\infty}e^{-qt}P_{t}f(x)dt,$$ where $(P_{t})_{t\geq 0}$ is the semigroup, $f$ is non negative ...
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Independent components of a $d-$dimesional Lévy process

I'm tryig to prove the next proposition but I'm lost: Let $X=(X^{1},\ldots,X^{d})$ be a $d-$dimensional Lévy process with Gaussian coefficient $Q$ and Lévy measure $\Pi.$ Then the Levy processes $X^1,...
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25 views

Doubt in the proof of existence and uniqueness of Laplace exponent for subordinators

I was reading a proof of the existence and uniqueness of Laplace exponent of subordinators: If $\Phi$ is the Laplace exponent of a subordinator,then there exist a unique pair $(k,d)$ of nonnegative ...
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44 views

Lévy measure as a limit

I'm reading about Lévy processes. Durig this, I've found with the next proposition without proof: For every fixed $a>0,$ the measure $\frac{1}{\epsilon}P_{0}(X_{\epsilon}\in dx)$ converges vaguely ...
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Comparing two sum of fractal moments for heavy-tail distribution

Assume a heavy tailed distribution whose tail can be approximated as $$P(X\geq x)\sim x^{-\alpha}$$ Consider some fractal moment of iid $X_i$, we have $$\frac{1}{n}\sum_{i=1}^nX_i^{\theta}\sim O(n^{\...
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First passage time $T_{B}^{'}=\inf\{t\geq0:X_{t}\in B\}$ is stopping time when $B$ is open or closed set

I'm reading a proof of the following proposition: Let $(X_{t})_{t\geq 0}$ be a Lévy process on $\mathbb{R}^{d}$ and $B\subset\mathbb{R}^{d}.$ We deine $T_{B}=\inf\{t>0:X_{t}\in B\}$ and $T_{B}^{'}=...
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196 views

Cauchy's Functional Equation

Consider Cauchy's Functional Equation $$\phi(t+s)=\phi(t)+\phi(s).$$ Can we say that any right continuous with left limits (cadlag) solution is Borel measurable? Obviously continuous solutions are ...
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$f(X_{T})=f(X_{T^{-}})$ a.s. for all $f \in C_0$ implies $X_{T}=X_{T^{-}}$ a.s.

During a proof of quasi-left continuity of Lévy processes, there is the next step: $$E(f(X_{T})g(X_{T}))=E(f(X_{T^{-}})g(X_{T}))$$ for all $f,g\in C_{0},$ where $C_{0}$ is the set of all continuous ...
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$(X_{z+t}-X_{z})_{t\geq 0}$ satisfies “Strong Markov Property” where $X$ is càdlag process and $z$ discrete stopping time.

I'm reading about strong Markov property. In the text there is the next proposition which I'm, trying to prove: If $X$ is a càdlag process and $z$ is a discrete stopping time, then $(X_{z+t}-X_{z})_{...
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Convergence of stopping times and limit of a right continuous process

I'm trying to prove the next: If $X$ is cád (right continuous) and adapted process, then $\displaystyle\lim_{n\rightarrow\infty}X_{Z_{n}}=X_{T}$ and $X_{T}$ is random variable. Here $T$ is a ...
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1answer
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convergence towards infinity of jumping times of Levy processes

The following is a remark of Philip Protter at page 26 of the book Stochastic integration and Differential equation that I have not been able to proved yet. Let $\Lambda$ be a borel set in $\mathbb{...
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139 views

Prove the following distribution is infinitely divisible

Consider the following characteristic function and show that the distribution is infinitely divisible $$\phi(u)=exp(imu-\sigma\vert u\vert[1+i\beta\frac{2}{\pi}\text{sgn}(u)\text{log}\vert u\vert])$$ ...
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1answer
54 views

Show that $ (e^{\alpha X_t} \int^ t_ 0 e ^{-\alpha X_u}du, t \geq 0) $ is a Markov process

I want to show that $ (e^{\alpha X_t} \int^ t_ 0 e ^{-\alpha X_u}du, t \geq 0) $ is a Markov process whereas $(\int^ t_ 0 e ^{-\alpha X_u}du, t\geq 0)$ is not. Here $X_t$ is a Levy Process and ...
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1answer
97 views

Give canonical decomposition of semimartingales $Z_t$ and $W_t$ based on $\mathscr{A}_t$ Levy's area

Let $(X_t, Y_t)$ be a two-dimensional $(\mathscr{F}_t)$-Brownian motion started from 0. We set, for every $t \geq 0$ $$\mathscr{A}_t = \int_0^t X_s dY_s - \int_0^t Y_s dX_s$$ (Levy's area). ...
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Fraction of the largest element of a sum of $N$ i.i.d. random variates sampled from power law distribution

For a probability distribution $$ p(x) \propto x^{-(\mu + 1)} \qquad 0 < \mu < 1$$ both the sum of $N$ i.i.d. samples $S_N$ and the largest element of those samples $x_{\text{max}}$ scale as $N^{...
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1answer
105 views

Generalizing a proof for the density of stopped subordinators

Is it possible to generalize the proof of the statement below to general Lévy kernels $\rho(r)dr$ or even Lévy-type kernels $\rho(x,r)dr$ (in the sense of Lévy Matters III by Böttcher et al.)? ...