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## SELF-SIMILARITY AND LAMPERTI CONVERGENCE FOR FAMILIES OF STOCHASTIC PROCESSES

Fonte: SPRINGER
Publicador: SPRINGER

Tipo: Artigo de Revista Científica

ENG

Relevância na Pesquisa

46.6%

#exponential tilting#fractional Hougaard motion#Hougaard Levy process#Lamperti transformation#power variance function#LEVY PROCESSES#Mathematics

We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to certain important families of processes that are not self-similar in the conventional sense. This includes Hougaard Levy processes such as the Poisson processes, Brownian motions with drift and the inverse Gaussian processes, and some new fractional Hougaard motions defined as moving averages of Hougaard Levy process. Such families have many properties in common with ordinary self-similar processes, including the form of their covariance functions, and the fact that they appear as limits in a Lamperti-type limit theorem for families of stochastic processes.; Danish Natural Science Research Council; FAPESP, Brazil

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## Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/03/2014

Relevância na Pesquisa

26.15%

We consider anisotropic self-similar random fields, in particular, the
fractional Brownian sheet. This Gaussian field is an extension of fractional
Brownian motion. We prove some properties of covariance function for
self-similar fields with rectangular increments. Using Lamperti transformation
we obtain properties of covariance function for the corresponding stationary
fields. We present an example of a Gaussian self-similar field with stationary
rectangular increments that is not a fractional Brownian sheet.

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## Optimal prediction for positive self-similar Markov processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/09/2014

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26.15%

This paper addresses the question of predicting when a positive self-similar
Markov process X attains its pathwise global supremum or infimum before hitting
zero for the first time (if it does at all). This problem has been studied in
Glover et al. (2013) under the assumption that X is a positive transient
diffusion. We extend their result to the class of positive self-similar Markov
processes by establishing a link to Baurdoux and van Schaik (2013), where the
same question is studied for a Levy process drifting to minus infinity. The
connection to Baurdoux and van Schaik (2013) relies on the so-called Lamperti
transformation which links the class of positive self-similar Markov processes
with that of Levy processes. Our approach will reveal that the results in
Glover et al. (2013) for Bessel processes can also be seen as a consequence of
self-similarity.

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## Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and processor-sharing queues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.15%

We study the convergence of the $M/G/1$ processor-sharing, queue length
process in the heavy traffic regime, in the finite variance case. To do so, we
combine results pertaining to L\'{e}vy processes, branching processes and
queuing theory. These results yield the convergence of long excursions of the
queue length processes, toward excursions obtained from those of some reflected
Brownian motion with drift, after taking the image of their local time process
by the Lamperti transformation. We also show, via excursion theoretic
arguments, that this entails the convergence of the entire processes to some
(other) reflected Brownian motion with drift. Along the way, we prove various
invariance principles for homogeneous, binary Crump-Mode-Jagers processes. In
the last section we discuss potential implications of the state space collapse
property, well known in the queuing literature, to branching processes.; Comment: Published in at http://dx.doi.org/10.1214/12-AAP904 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## Exact and asymptotic $n$-tuple laws at first and last passage

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.44%

Understanding the space-time features of how a L\'evy process crosses a
constant barrier for the first time, and indeed the last time, is a problem
which is central to many models in applied probability such as queueing theory,
financial and actuarial mathematics, optimal stopping problems, the theory of
branching processes to name but a few. In \cite{KD} a new quintuple law was
established for a general L\'evy process at first passage below a fixed level.
In this article we use the quintuple law to establish a family of related joint
laws, which we call $n$-tuple laws, for L\'evy processes, L\'evy processes
conditioned to stay positive and positive self-similar Markov processes at both
first and last passage over a fixed level. Here the integer $n$ typically
ranges from three to seven. Moreover, we look at asymptotic overshoot and
undershoot distributions and relate them to overshoot and undershoot
distributions of positive self-similar Markov processes issued from the origin.
Although the relation between the $n$-tuple laws for L\'evy processes and
positive self-similar Markov processes are straightforward thanks to the
Lamperti transformation, by inter-playing the role of a (conditioned) stable
processes as both a (conditioned) L\'evy processes and a positive self-similar
Markov processes...

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## Breadth first search coding of multitype forests with application to Lamperti representation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/10/2014

Relevância na Pesquisa

26.58%

We obtain a bijection between some set of multidimensional sequences and this
of $d$-type plane forests which is based on the breadth first search algorithm.
This coding sequence is related to the sequence of population sizes indexed by
the generations, through a Lamperti type transformation. The same
transformation in then obtained in continuous time for multitype branching
processes with discrete values. We show that any such process can be obtained
from a $d^2$ dimensional compound Poisson process time changed by some integral
functional. Our proof bears on the discretisation of branching forests with
edge lengths.

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## Hitting properties and non-uniqueness for SDE driven by stable processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.28%

We study a class of self-similar jump type SDEs driven by H\"older-continuous
drift and noise coefficients. Using the Lamperti transformation for positive
self-similar Markov processes we obtain a necessary and sufficient condition
for almost sure extinction in finite time. We then show that for certain
parameters pathwise uniqueness holds in a restricted sense, namely among
solutions spending a Lebesgue-negligible amount of time at 0. A direct power
transformation plays a key role.

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## Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/06/2008

Relevância na Pesquisa

26.15%

Two types of Gaussian processes, namely the Gaussian field with generalized
Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy
covariance (GSGCC) are considered. Some of the basic properties and the
asymptotic properties of the spectral densities of these random fields are
studied. The associated self-similar random fields obtained by applying the
Lamperti transformation to GFGCC and GSGCC are studied.; Comment: 32 pages, 6 figures

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## Deep factorisation of the stable process

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

16.83%

The Lamperti--Kiu transformation for real-valued self-similar Markov
processes (rssMp) states that, associated to each rssMp via a space-time
transformation, there is a Markov additive process (MAP). In the case that the
rssMp is taken to be an $\alpha$-stable process with $\alpha\in(0,2)$, Chaumont
et al. (2013) and Kuznetsov et al. (2014) have computed explicitly the
characteristics of the matrix exponent of the semi-group of the embedded MAP,
which we henceforth refer to as the {\it Lamperti-stable MAP}. Specifically,
the matrix exponent of the Lamperti-stable MAP's transition semi-group can be
written in a compact form using only gamma functions.
Just as with L\'evy processes, there exists a factorisation of the (matrix)
exponents of MAPs, with each of the two factors uniquely characterising the
ascending and descending ladder processes, which themselves are again MAPs.
To the author's knowledge, not a single example of such a factorisation
currently exists in the literature. In this article we provide a completely
explicit Wiener--Hopf factorisation for the Lamperti-stable MAP. As a
consequence of our methodology, we also get additional new results concerning
space-time invariance properties of stable processes. Accordingly we develop
some new fluctuation identities therewith.

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## Proof(s) of the Lamperti representation of Continuous-State Branching Processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

36.6%

This paper uses two new ingredients, namely stochastic differential equations
satisfied by continuous-state branching processes (CSBPs), and a topology under
which the Lamperti transformation is continuous, in order to provide
self-contained proofs of Lamperti's 1967 representation of CSBPs in terms of
spectrally positive L\'evy processes. The first proof is a direct probabilistic
proof, and the second one uses approximations by discrete processes, for which
the Lamperti representation is evident.

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## On continuous state branching processes: conditioning and self-similarity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/12/2007

Relevância na Pesquisa

26.58%

In this paper, for $\alpha\in (1, 2}$ we show that the $\alpha$-stable
continuous-state branching process and the associated process conditioned never
to become extinct are positive self-similar Markov processes. Understanding the
interaction of the Lamperti transformation for continuous-state branching
processes and the Lamperti transformation for positive self-similar Markov
processes permits accessto a number of explicit results concerning the paths of
stable-continuous branching processes and its conditioned version.

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## Non-homogeneous random walks on a half strip with generalized Lamperti drifts

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/12/2015

Relevância na Pesquisa

26.81%

We study a Markov chain on $\mathbb{R}_+ \times S$, where $\mathbb{R}_+$ is
the non-negative real numbers and $S$ is a finite set, in which when the
$\mathbb{R}_+$-coordinate is large, the $S$-coordinate of the process is
approximately Markov with stationary distribution $\pi_i$ on $S$. If $\mu_i(x)$
is the mean drift of the $\mathbb{R}_+$-coordinate of the process at $(x,i) \in
\mathbb{R}_+ \times S$, we study the case where $\sum_{i} \pi_i \mu_i (x) \to
0$, which is the critical regime for the recurrence-transience phase
transition. If $\mu_i(x) \to 0$ for all $i$, it is natural to study the
\emph{Lamperti} case where $\mu_i(x) = O(1/x)$; in that case the recurrence
classification is known, but we prove new results on existence and
non-existence of moments of return times. If $\mu_i (x) \to d_i$ for $d_i \neq
0$ for at least some $i$, then it is natural to study the \emph{generalized
Lamperti} case where $\mu_i (x) = d_i + O (1/x)$. By exploiting a
transformation which maps the generalized Lamperti case to the Lamperti case,
we obtain a recurrence classification and existence of moments results for the
former. The generalized Lamperti case is seen to be more subtle, as the
recurrence classification depends on correlation terms between the two
coordinates of the process.; Comment: 19 pages

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## A Jump Type SDE Approach to Positive Self-Similar Markov Processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

16.15%

We present a new approach to positive self-similar Markov processes (pssMps)
by reformulating Lamperti's transformation via jump type SDEs. As applications,
we give direct constructions of pssMps (re)started continuously at zero if the
Lamperti transformed Levy process is spectrally negative. Our paper can be seen
as a continuation of similar studies for continuous state branching processes
but the approach seems to be more fruitful in the context of pssMps.; Comment: 38 pages

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## Representation of self-similar Gaussian processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.15%

We develop the canonical Volterra representation for a self-similar Gaussian
process by using the Lamperti transformation of the corresponding stationary
Gaussian process, where this latter one admits a canonical integral
representation under the assumption of pure non-determinism. We apply the
representation obtained for the self-similar Gaussian process to derive an
expression for Gaussian processes that are equivalent in law to the
self-similar Gaussian process in question.

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