# Questions tagged [levy-processes]

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

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### Proof that the jump measure of a Lévy process is a Poisson random measure

Let $(X_t)_{t \geq 0}$ be an $\mathbb{R}^d$-valued Lévy process and consider its associated jump measure $N_t: \Omega \times \mathbb{B}(\mathbb{R}^d \setminus \{0\}) \to \bar{\mathbb{N}}_0$ given by \...
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### Why does the infinitesimal of a poisson process behave as it does in the Ito multiplication table

In more informal derivations of Ito's formula for jump processes, the multiplication table for $dt, dW_t$ and $dN_t$give that $dN_tdN_t=dN_t$. Why is this? I have tried deriving this from the ...
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### Q: Levy process as a combination of wiener process poisson procerss and determenistic process

Is it true that any Levy process can be decomposed to a 3 processes: 1. Poisson process 2. Wiener process 3. determenistic process If so, what is the proof/intuitaion for that? thank you very much
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### Path of Excursion Process

I am learning the excursion theory for Markov Process and have some difficulties to visualize the excursion process $\epsilon_t$ of a Markov Process $X_t$. Based on my understanding of the concept (...
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### Proof Levy Khintchine

I am supposed to proof this theorem Theorem 1.16 For a semimartingale levy process X, there is a continuous non decreasing function $\beta$ with $\beta_0 = 0$, a measure $\kappa$ satisfying the ...
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### Two questions about densities of a Levy process

I have the following two questions about densities of a Levy process $(X_t)_{t\ge 0}$. Does $X_t$ have a density wrt. Lebesque measure? If not, under what conditions does this hold? If it does have ...
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### Independence of increments with stopping times in Levy processes

Let $X$ be a Levy Process and $S<T<U<V$ be stopping times. Let $F^X$ be the natural filtration of $X$. How can one show that $X_V - X_U$ and $X_T - X_S$ are independent and $X_V - X_U$ ...
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### subordinator, which process has this levy measure

which subordinator has this levy measure $v_Z(dx)={1}_{x>0}\gamma x^{-a-1}e^{-\lambda x},dx(a\in [0,1),\gamma,\lambda>0)$ with positive drift $\sigma_Z$. Can I find this in any book? I suppose ...
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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 $$X_t=\sigma B_t + \sum_{i=1}^{N_t}\eta_i(X_t)$$ a Levy process? Here $B_t$ is a standard Brownian motion and $N_t$ is a ...
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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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### 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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### 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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### 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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### 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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### 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 ...
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 $... 0answers 331 views ### Laplace exponent of a standard$\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$is ... 1answer 49 views ### Does the Levy process stay in any open ball at any fixed time with positive probability? Question: Let$X$be a$d$-dimensional Levy process. Then for every$t>0$and$a>0$, \begin{equation}\tag{1} \mathbf P\{|X_t|<a\}>0\ ? \end{equation} The question comes from the proof ... 1answer 47 views ### Recurrent Lévy process implies$\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 $$\space\displaystyle\limsup_{t\rightarrow\infty}X_{t}=\infty\space\text{a.s.}\space\text{and}\space\displaystyle\... 1answer 63 views ### Asymptotic behavior of Lévy processes I'm trying to prove the next proposition: Let \{X_{t}\}_{t\geq 0} be a non-zero Lévy process on \mathbb{R}. Then it satisfies one of the following three conditions:$$i)\space\displaystyle\lim_{t\... 2answers 37 views ### 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$\...
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 ...