# 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 $$X_t=\sigma B_t + \sum_{i=1}^{N_t}\eta_i(X_t)$$ 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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### 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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### 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 ...
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### 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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### 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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