Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.
Questions tagged [levy-processes]
384 questions
12
votes
1 answer
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
- 10,622
- 6
- 49
- 122
12
votes
0 answers
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
- 123,507
8
votes
1 answer
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
- 808
7
votes
0 answers
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…
parsiad
- 25,738
7
votes
1 answer
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
- 10,622
- 6
- 49
- 122
7
votes
2 answers
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
- 10,622
- 6
- 49
- 122
7
votes
1 answer
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
- 3,796
6
votes
1 answer
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…
BigMike
- 361
6
votes
2 answers
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
$$(H\star M)_t:=\int_0^t H_{s}dM_s=\int_0^t…
Pasriv
- 1,120
6
votes
2 answers
Central Limit Theorem for Lévy Process
I am reading a book, which uses the Central Limit Theorem of Lévy Processes $X_t$ without mentioning the exact theorem.
Due to the infinite divisible property I can write $X_t$ as a sum of $N$ iid random variables $X^i$
$$
X_t=\sum_{i=1}^N…
ziT
- 637
6
votes
0 answers
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
- 908
5
votes
1 answer
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,…
Sasha
- 71,686
5
votes
1 answer
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
- 51
5
votes
1 answer
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
- 544
5
votes
1 answer
What is the true definition of a Lévy process?
What is the “true” definition of a Lévy process?
I notice that definitions vary in non-equivalent ways:
1) Wikipedia states that a Lévy process is one that satisfies four particular properties, but these properties do not include the…
Michael
- 26,378