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SELF-SIMILARITY AND LAMPERTI CONVERGENCE FOR FAMILIES OF STOCHASTIC PROCESSES

JORGENSEN, Bent; MARTINEZ, Jose R.; DEMETRIO, Clarice G. B.
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
46.6%
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

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

Makogin, Vitalii; Mishura, Yuliya
Tipo: Artigo de Revista Científica
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.

Optimal prediction for positive self-similar Markov processes

Baurdoux, Erik; Kyprianou, Andreas; Ott, Curdin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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.

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

Lambert, Amaury; Simatos, Florian; Zwart, Bert
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)

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

Kyprianou, A.; Pardo, J. C.; Rivero, V.
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...

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

Chaumont, Loïc
Tipo: Artigo de Revista Científica
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.

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

Berestycki, Julien; Doering, Leif; Mytnik, Leonid; Zambotti, Lorenzo
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.

Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Lim, S. C.; Teo, L. P.
Tipo: Artigo de Revista Científica
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

Deep factorisation of the stable process

Kyprianou, Andreas E.
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.

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

Caballero, Maria-Emilia; Lambert, Amaury; Bravo, Geronimo Uribe
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.

On continuous state branching processes: conditioning and self-similarity

Kyprianou, A. E.; Pardo, J. C.
Tipo: Artigo de Revista Científica
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.

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

Lo, Chak Hei; Wade, Andrew R.
Tipo: Artigo de Revista Científica
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

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

Doering, Leif; Barczy, Matyas