# Questions tagged [levy-processes]

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

215 questions
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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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### 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 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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### 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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### 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 ...
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 $X^1,... 0answers 25 views ### Doubt in the proof of existence and uniqueness of Laplace exponent for subordinators I was reading a proof of the existence and uniqueness of Laplace exponent of subordinators: If$\Phi$is the Laplace exponent of a subordinator,then there exist a unique pair$(k,d)$of nonnegative ... 0answers 44 views ### Lévy measure as a limit I'm reading about Lévy processes. Durig this, I've found with the next proposition without proof: For every fixed$a>0,$the measure$\frac{1}{\epsilon}P_{0}(X_{\epsilon}\in dx)$converges vaguely ... 0answers 20 views ### 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 $$\frac{1}{n}\sum_{i=1}^nX_i^{\theta}\sim O(n^{\... 0answers 33 views ### First passage time T_{B}^{'}=\inf\{t\geq0:X_{t}\in B\} is stopping time when B is open or closed set I'm reading a proof of the following proposition: Let (X_{t})_{t\geq 0} be a Lévy process on \mathbb{R}^{d} and B\subset\mathbb{R}^{d}. We deine T_{B}=\inf\{t>0:X_{t}\in B\} and T_{B}^{'}=... 1answer 196 views ### Cauchy's Functional Equation Consider Cauchy's Functional Equation$$\phi(t+s)=\phi(t)+\phi(s).$$Can we say that any right continuous with left limits (cadlag) solution is Borel measurable? Obviously continuous solutions are ... 1answer 33 views ### f(X_{T})=f(X_{T^{-}}) a.s. for all f \in C_0 implies X_{T}=X_{T^{-}} a.s. During a proof of quasi-left continuity of Lévy processes, there is the next step:$$E(f(X_{T})g(X_{T}))=E(f(X_{T^{-}})g(X_{T}))$$for all f,g\in C_{0}, where C_{0} is the set of all continuous ... 0answers 26 views ### (X_{z+t}-X_{z})_{t\geq 0} satisfies “Strong Markov Property” where X is càdlag process and z discrete stopping time. I'm reading about strong Markov property. In the text there is the next proposition which I'm, trying to prove: If X is a càdlag process and z is a discrete stopping time, then (X_{z+t}-X_{z})_{... 0answers 41 views ### Convergence of stopping times and limit of a right continuous process I'm trying to prove the next: If X is cád (right continuous) and adapted process, then \displaystyle\lim_{n\rightarrow\infty}X_{Z_{n}}=X_{T} and X_{T} is random variable. Here T is a ... 1answer 63 views ### convergence towards infinity of jumping times of Levy processes The following is a remark of Philip Protter at page 26 of the book Stochastic integration and Differential equation that I have not been able to proved yet. Let \Lambda be a borel set in \mathbb{... 0answers 139 views ### Prove the following distribution is infinitely divisible Consider the following characteristic function and show that the distribution is infinitely divisible$$\phi(u)=exp(imu-\sigma\vert u\vert[1+i\beta\frac{2}{\pi}\text{sgn}(u)\text{log}\vert u\vert])$$... 1answer 54 views ### Show that (e^{\alpha X_t} \int^ t_ 0 e ^{-\alpha X_u}du, t \geq 0) is a Markov process I want to show that (e^{\alpha X_t} \int^ t_ 0 e ^{-\alpha X_u}du, t \geq 0) is a Markov process whereas (\int^ t_ 0 e ^{-\alpha X_u}du, t\geq 0) is not. Here X_t is a Levy Process and ... 1answer 97 views ### Give canonical decomposition of semimartingales Z_t and W_t based on \mathscr{A}_t Levy's area Let (X_t, Y_t) be a two-dimensional (\mathscr{F}_t)-Brownian motion started from 0. We set, for every t \geq 0$$\mathscr{A}_t = \int_0^t X_s dY_s - \int_0^t Y_s dX_s$$(Levy's area). ... 0answers 22 views ### Fraction of the largest element of a sum of N i.i.d. random variates sampled from power law distribution For a probability distribution$$ 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^{...
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.)? ...