Mathematics

Questions tagged [levy-processes]

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

389 questions
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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 jumps of value 1. It is almost surely increasing. When…
user119615
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Holomorphic extensions of characteristic functions

Let $\chi(\xi) := \mathbb{E}e^{i \xi X}$ , $\xi \in \mathbb{R}^d$ , be the characteristic function of a random variable $X$ . It is widely known that $\chi$ admits a holomorphic extension to the strip $\{z \in \mathbb{C}; |\text{Im} \, z| < m\}$…
saz
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The Lévy-Khintchine formula and integrability conditions of a random measure

I am trying to see the connection between the Lévy-Khintchine and the integrability conditions of a Lévy measure. The literature seems to always connect both, but I cannot make sense of this relation nor found a proof. Perhaps somebody can give me…
Adam
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Inequality for Lévy SDE

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$…
parsiad
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When is the compensated Poisson random measure a martingale ? (extensions to sets not bounded from 0)

Assume you have a Lévy process X. Let $N(t,A)$ be defined as the number of jumps in the interval $(0,t]$, such that the jumps size $\Delta X_s \in A$. It can be shown that if $0 \ne \bar{A}$, then $N(t,A)$ is a Poisson process with intensity…
user119615
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Good book that contains stochastic integration, martingales and Lévy-processes?

Does anyone know about any good and easy interoductory books which contins information about martingales, sotchastic integration and Lévy-processes? I have tried reading:…
user119615
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Stochastic Integration with respect to Cauchy Process?

I'm interested in a one-dimensional stochastic process: $$dX_t = f(X_t)dt + g(X_t) dZ_t$$ where $Z_t$ is a Cauchy process and $f,g$ are nice polynomials (I'm looking at an ODE that gets perturbed by noise but where the noise has large tails). $Z_t$…
Brenton
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Law of large numbers for a Subordinator.

Let $\left( X_{t}\right) _{t\geq0}$ be a subordinator with the Laplace exponent given by $$ \Phi\left( \lambda\right) =d\lambda+\int_{0}^{\infty}\left( 1-e^{-\lambda x}\right) \nu\left( dx\right) $$ Show that almost surely $$…
BigMike
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Stochastic Integral with respect to Compensated Poisson Process

Proposition : Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral $$(H\star M)_t:=\int_0^t H_{s}dM_s=\int_0^t…
Pasriv
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Central Limit Theorem for Lévy Process

I am reading a book, which uses the Central Limit Theorem of Lévy Processes $X_t$ without mentioning the exact theorem. Due to the infinite divisible property I can write $X_t$ as a sum of $N$ iid random variables $X^i$ $$ X_t=\sum_{i=1}^N X^i_{t/N}…
ziT
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What is the Lévy measure of the Student's $t$-distribution?

It is known since the 1970's that the Student's $t$-distribution is infinitely divisible. We can therefore apply the Lévy-Khintchine representation to it, and define the Lévy measure associated to a Student $t$-distribution. Question: What is known…
Goulifet
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Computing quadratic variation for stable Levy flights with $0<\alpha<2$?

The wiki page on semi-martingales states that Every Lévy process is a semimartingale. and that The quadratic variation exists for every semimartingale. Let $X_t$ be a stable Levy process with $X_t$ distributed as $S(\alpha, \beta, \mu \, t, \sigma…
Sasha
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Does Weak Convergence plus a Uniform Moment Bound Imply Uniform Integrability for Lévy Measures?

Let $\{X_n\}_{n \ge 1}$ be a sequence of $p$ -dimensional infinitely divisible (ID) random vectors, and let $\{\mu_n\}_{n \ge 1}$ be their corresponding Lévy measures. Suppose we have two conditions: The sequence of random vectors converges weakly…
Gregório Vieira
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Stochastic continuity in the definition of a Lévy process

A Lévy process is defined as a stochastic process, $X = (X_t)_{t\geq 0}$ , with the properties: L1. $X_0 = 0$ a.s. L2. $X$ has independent increments L3. $X$ has stationary increments L4. $X$ is continuous in probability What is the purpose of…
piers
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Show that a Levy measure $\nu$ (which arises from a convergence of Infinitely Divisible random vectors) is such that $\int x d\nu(x)=0$

Let $(X_{jn})_{1\leq j \leq n}$ be a triangular array of $p-$ dimensional random vectors (row independent). Suppose $X_{jn} \sim \mu_{jn}$ and 1. $\,\, E X_{jn}= \int_{\mathbb R^p} x d \mu_{jn}=0$ 2. $\,\,\lim_{n \to \infty} \max_{1\leq j \leq n}…
PSE
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