Questions tagged [levy-processes]
Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.
349
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Level-crossing time of Levy process is larger than a fixed time with positive probability
Let $(X_t)_{t\geq 0}$ be a Lévy process on $\Bbb R$ with characteristic triple $(a, \sigma , \pi)$. For a level $K>0$ define
$$\tau_K := \inf\{t>0 : X_t > K\} $$
Question: When does it hold ...
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1
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PDF of stable distribution in terms of the Fox $H$-function for the case $\alpha > 1$.
The PDF of Stable Distribution in terms of a Fox H-function for the case $α≤1$ is available at https://reference.wolfram.com/language/ref/FoxH.html (Examples\Applications) in the form:
$S(x;\alpha,\...
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1
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Expectation of waiting time ratios
Question:
At a restaurant's bathroom, men and women arrive at the same rate, on average 20 minutes. Assume the number of people who waiting in the queue is Poisson distributed. Men spend on average 60 ...
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Hamilton-Jacobi-Bellman equation for Levy processes
I am trying to understand a optimal investment/stochastic control Problem and derive the HJB equation for following Wealth Process
$dX^{\phi}(t)=\int_{0}^{t} X^{\phi}(s-)(r+\phi(s)(\mu-r))ds+\int_{0}^{...
- 51
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Domain of the infinitesimal generator of subordinators
Let $(X_t)$ be a subordinator (not killed). Since $(X_t)$ is a non-decreasing Levy process, we have the corresponding infinitesimal generator:
\begin{equation}
Af(x)=\delta f'(x)+\int_{0}^{\infty}(f(x+...
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30
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What is the cardinality of the number of jumps in a Lévy process on a given time interval?
If a Lévy process can be defined as a right continuous process with existing left limits, then according to the following linked theorem (Prove that the number of jump discontinuities is countable for ...
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Compound Poisson Process infinitely many jumps?
When I look at the following compound Poisson Process, where $(N_{t})_{t\geq0}$ is a Poisson Process with Parameter $\lambda$ and $\xi_{i}$ are $\mathbb{R}$ valued i.i.d random Variables independent ...
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Interpret conditional expextation of stochastic process
I am trying to understand what the follwing means
$$ v(x) = \sup_\tau \mathbb{E}_x(f(X_\tau)) = \sup_\tau \mathbb{E}(f(X_\tau) | X_0 = x) = \sup_\tau \frac{\mathbb{E}(1_{X_0 =x} f(X_\tau))}{P(X_0 = x)}...
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Exercise on Levy measure
I have to prove that a Borel measure $\mu$ on $\mathbb{R}^+$ such that
$$
\int_0^\infty (z\wedge z^2)\mu(dz)<\infty
$$
is a Levy measure.
MY ATTEMPT:
We need to check that $\int_0^\infty(1\wedge z^...
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Poisson Process/Lévy process
I want to prove that a Poisson process fulfils the continuity in probability property (https://en.wikipedia.org/wiki/L%C3%A9vy_process#Mathematical_definition)
$lim_{h \rightarrow 0} \mathbb{P}(|N_{t+...
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Finding finite dimensional distribution of Levy process from the characteristic function
This is part of a statement from From Levy type processes to Parabolic SPDEs by Rene SChilling.
Corollary 2.4 states that the finite dimensional distributions $P(X_{t_1}\in dx_1 ,\dots, X_{t_n}\in ...
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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....
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Show that a Levy measure (which arises from a convergence of Infinitely Divisible random vectors) has infinite total mass
Given an array of probability measures $(\mu_{jn})_{1\leq j \leq n}$, all defined on borelians of $\mathbb R^p$. Let $S_n'$ a sequence of $p-$dimensional random vectors with $S_n' \Longrightarrow X$ ...
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Fractional integral related to stable process on half-space
I'm working with the isotropic $\alpha$-stable Lévy process in $\mathbb{R}^d$ $(\alpha \in (0,1) \text{ and } d \geq 2)$. I know that the distribution of the first hitting of this process into the ...
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If $\nu$ is a Levy measure, how to show that $\nu( (-\epsilon,\epsilon)^c ) <\infty$
We know that the Levy measure on borelians is a measure $\nu$ such that
\begin{equation}
\int \frac{|x|^2}{1+ |x|^2} \nu (dx) < \infty
\end{equation}
We can show that this is equivalent to:
\...
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1
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Example of a Levy measure with infinite mass around zero but with finite second moment
Let $X$ be an infinitely divisible random variable with the "Lévy–Khintchine triplet" $(a,\sigma, \nu)$ representation where $d\nu (x)=\frac 1 {x^{\alpha}} \chi_{(0,\infty)}dx$ with $ 1<\...
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Example of a Levy measure with infinite mass around zero
A measure $\nu$ is a Levy measure if:
\begin{equation}
\nu(\{0\}),\quad \int_{|x|<1}|x|^{2} \nu (dx) <\infty, \quad \int_{|x| \geq 1} \nu (dx)<\infty.
\end{equation}
At first, I ...
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Equivalence of two integral constraints
If $\lambda$ is a Levy measure how can you show that the condition
$$
\int_0^\infty (y\wedge1)\;\lambda (dy) < \infty
$$
is equivalent to
$$
\int_0^\infty\frac y {1+y} \;\lambda (dy) < \infty.
$$...
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Running Maximum of a Levy process has the same distribution as the running maximum of the negative process?
I came across the following statement:
Let $X_t$ be a Levy process, $\overline{X}_t$ be the running maximum and $X_t^+= \max\{X_t, 0\}$. Then $\overline{X}_t - X_t^+ = \min\{\overline{X}_t, \overline{...
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How do I derive the law of supremum of a generic Levy process?
Let ${\bar x} > 0 $, $\sigma >0$, $\mu \in {\mathbb R}$, $T > 0 $ and $\alpha \in {\mathbb R}$.
Let $\phi(x) := \exp(-x^2/2)/\sqrt{2\pi} $ be a probability density of a standard normal ...
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Stochastically continuous implies existence càdlàg modiciation?
(From the book 'Lévy processes and infinitely divisble distributions, Ken-Iti Sato, 1999', page 59/60)\
My question:
Where exactly has Lemma 11.2 been used in the proof of lemma 11.3?
Lemma 11.2: $\...
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Proof for the generalized moments of Levy-Process.
I am struggling with the proof of the theorem for generalized moments (25.3) from Ken-Iti-Sato, 'Levy Processes and Infinitely divisible distributions'. Almost in the end of the proof we try to reduce ...
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How to build a Poisson random measure according a Levy Process
Let $(\Omega, \mathcal{F}, P)$ and $(\Theta, \mathcal{B}, \rho)$. A Poisson random measure (PRM) with intensity $\rho$ is a kernel $\mathcal{N}: \Omega \times \mathcal{B} \to \mathbb{R}$ such that:
$\...
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Time Laplace transform for Lévy process?
For which Lévy processes do I know the time Laplace transform? So if $p(t,x)$ is the density function of the process, when do I know
\begin{align*}
\int_0^\infty e^{-\lambda t} p(t,x) dt, \quad \...
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Deriving the Lévy triplet of homogeneous Poisson processes.
How would one derive the Lévy triplet of a time-homogeneous Poisson process? My idea is to make a direct comparison between the characteristic function of a Poisson process and the characteristic ...
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answers
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Lévy process, indicator function, sojourn time
Given a one dimensional Lévy process $X_t$ with characteristic exponent $\psi(\xi)$, so that
\begin{align}
\mathbf{E}[e^{i \xi X_t}]= e^{t \psi(\xi)}
\end{align}
for example we can find $\psi(\xi)$ ...
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Can be a Cox process or a Compund Cox process expressible as Lévy-Itô stochastic integral?
The (inhomogeneous) Poisson process is stochastic process whose Lagrange functional is given by:
$$ L_N(f) = \exp\left(-\int_{\textbf{R}}(1-e^{-f(x)})\Lambda(\mathrm{d}x)\right) $$
being $\Lambda(\...
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Levy process "that has only jumps"
I post this today as I'm looking to some caractérisation of a Levy processes "that has only jumps" but I didn't found anything on the web neither on the classic books I know about the ...
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1
answer
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Why is $-cN/z^\alpha$ a good approximation for $\ln(1-F(z))^N$?
Bertin's Statistical Physics of Complex Systems, 3rd ed. p. 65 defines a "complementary cumulative distribution $\tilde{F}(z)$" equal to $\int_z^\infty p(x)\,dx$, where the density $p(z)$ is ...
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Equivalence of a Levy measure
We know that a measure $\nu$ is a Levy measure if:
\begin{equation}\label{1}\tag{1}
\nu(\{0\}),\quad \int_{|x|<1}|x|^{2} \nu (dx) <\infty, \quad \int_{|x| \geq 1} \nu (dx)<\infty.
\end{...
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When are levy flights superdiffusive and why? [closed]
A stable distribution that is centered and symmetrical has a value $\alpha \leq 1$ the expected value of the distribution is undefined.
I have also been told that if a Levy Flight has it's step ...
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Lévy process, characteristic function, $\frac{\psi(\xi)}{i \xi}$
Given a one dimensional Lévy process $X_t$ with characteristic exponent $\psi(\xi)$, so that
\begin{align}
\mathbf{E}[e^{i \xi X_t}]= e^{t \psi(\xi)}
\end{align}
for example we can find $\psi(\xi)$ ...
- 79
1
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0
answers
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From Donsker's theorem to subordinate Brownian motion
Denote $S_n := \sum_{k=1}^n X_i$, where the random variables $X_i$ are independent and identically distributed. Suppose $\mathbb{E}[X_i] = \mu$ and $\mathbb{V}[X_i] = \sigma^2$ are finite. Let $\...
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Suppose $\xi_t$ an infinitely divisible cadlag process with Lévy measure $\Lambda((-\infty,\,0])=0$. All jumps nonnegative?
Suppose $\xi_t$ defined on $[0,\,\infty)$ with $\xi_t(\omega)$ always nonnegative, is an infinitely divisible cadlag process whose Lévy measure $\Lambda$ concentrates on $(0,\,\infty)$. Is this ...
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Jumps of a compound Poisson process
If $(Z_n)_{n\in\mathbb N}$ is an i.i.d. process with values in a normed $\mathbb R$-vector space $E$, then $$W_n:=\sum_{i=1}^nZ_i$$ is called random walk with step distribution $\mathcal L(Z_1)$.
Now ...
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Does $\int\min(1,\|x\|)<\infty$ imply $\int\left|e^{{\rm i}\langle x,\:x'\rangle}-1-{\rm i}\langle x,x'\rangle1_{\overline B_1(0)}(x)\right|<\infty$
Let $E$ be a normed $\mathbb R$-vector space and $\lambda$ be a measure on $\mathcal B(E)$ with $$\int\min(1,\|x\|_E)\:\lambda({\rm d}x)<\infty.\tag1$$ Are we able to conclude $$\int\left|\...
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Lévy characterization of Hilbert-space-valued Wiener process
Let $H$ be a $\mathbb R$-Hilbert space (assume $H=\mathbb R^d$ for some $d\in\mathbb N$, if this is helpful for you to understand the following) and $(X_t)_{t\ge0}$ be an $H$-valued continuous Lévy ...
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vote
1
answer
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Show that every continuous Lévy process is a Wiener process
Let $H$ be a $\mathbb R$-Hilbert space (assume $H=\mathbb R^d$ for some $d\in\mathbb N$, if this is helpful for you to understand the following) and $(X_t)_{t\ge0}$ be an $H$-valued continuous Lévy ...
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If $X$ is a Lévy process, then $\operatorname{Var}[X_n]=n\operatorname{Var}[X_1]$
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $H$ be a $\mathbb R$-Hilbert space, $(X_t)_{t\ge0}$ be an $H$-valued Lévy process on $(\Omega,\mathcal A,\operatorname P)$ and $\mu_t:=...
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Is the jump measure of a space homogeneous Markov process translation invariant?
Let $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued Lévy process. $X$ is a space- and time-homogeneous Markov process with transition semigroup $$\kappa_t(x,B):=\...
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Is the limit in probability of Lévy processes still a Lévy process?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ bea filtration on $(\Omega,\mathcal A)$ and $(X_t)_{t\ge0}$ be a process on $(\Omega,\mathcal A,\operatorname ...
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1
vote
1
answer
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Bound for the probability of a Lévy process having a jump of a given size
Let
$E$ be a normed $\mathbb R$-vector space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$...
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2
votes
0
answers
28
views
If $M,N$ are martingales, show that $\operatorname E\left[M_tN_t\right]=\operatorname E\left[\sum_{s\in(0,\:t]}\Delta M_s\Delta N_s\right]$
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ and $M,N\in\mathcal V$ (see definition below$^2$) be càdlàg $\...
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2
votes
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answers
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Creeping for a Levy processes with infinite total mass
I am reading Theorem 7.11 from the book on Levy processes by Kyprianoy. I cannot intuitively grasp this:
In this theorem it is said that if the process has a Gaussian component then it creeps upwards. ...
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votes
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answer
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If $\pi_t$ is the jump measure of a Lévy process and $B_1,\ldots,B_k$ are disjoint, then $\pi_t(B_1),\pi_t(B_k)$ are independent
Let
$E$ be a normed $\mathbb R$-vector space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$...
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1
vote
0
answers
16
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If $r:=\operatorname{dist}(0,\overline B)$, what's the relation between $\{t>τ^B_{n-1}:\Delta x(t)\in B\}$ and $\{s\in(0,t]:\|\Delta x(s)\|_E\ge r\}$
Let
$E$ be a normed $\mathbb R$-vector space
$x:[0,\infty)\to E$ be càdlàg
$x(t-):=\lim_{s\to t-}x(s)$ and $\Delta x(t):=x(t)-x(t-)$ for $t\ge0$
$\tau^B_0:=0$, $I^B_n:=\{t>\tau^B_{n-1}:\Delta x(t)\...
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1
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0
answers
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Show that the inter-jump times of a Lévy process are i.i.d.
Let $E$ be a normed $\mathbb R$-vector space, $(X_t)_{t\ge0}$ be a càdlàg Lévy process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$, $B\in\mathcal B(E)$ ...
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vote
0
answers
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Show that $\sum_{\substack{s\ge0\\\Delta X_s\ne0}}1_B(s,\Delta X_s)$ is measurable
Let $E$ be a normed $\mathbb R$-vector space and $x:[0,\infty)\to E$ be regular in the sense that $$x(t\pm):=\lim_{s\to t\pm}x(s)$$ exist for all $t\ge0$. Let $$\Delta x(t):=x(t+)-x(t-)\;\;\;\text{for ...
- 13.5k
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answers
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Is the reciprocal of Lévy-Khintchine theorem for Levy process true?
We know that if $X = [X_t , t \in \mathbb{Z}]$ is a Levy process, then any marginal distribution has the charasteristic function - ch. f. of $X_t$ - given by $\varphi_t(r)=e^{t \phi(r)}$ whith:
$$\...
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votes
0
answers
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Distribution of first jump time
Let $(X_t)_{t\ge0}$ be a Lévy process, $\tau_0:=0$ and $$\tau_n:=\{t>\tau_{n-1}:\Delta X_t\in B\}\;\;\;\text{for }n\in\mathbb N$$ for some measurable set $B$ with $0\not\in B$.
How can we show that ...
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