arXiv subjects

Juan Carlos Pardo

Publications and source records attributed to Juan Carlos Pardo.

At least 37 records · Page 2Linked to original sources

arXiv2020

The Wright-Fisher model with efficiency

In populations competing for resources, it is natural to ask whether consuming fewer resources provides any selective advantage. To answer this question, we propose a Wright- Fisher model with two types of individuals: the inefficient individuals, those who need more resources to reproduce and can have a higher growth rate, and the efficient individuals. In this model, the total amount of resource N, is fixed, and the population size varies randomly depending on the number of efficient individuals. We show that, as N increases, the frequency process of efficient individuals converges to a diffusion which is a generalisation of the Wright- Fisher diffusion with selection. The genealogy of this model is given by a branching-coalescing process that we call the Ancestral Selection/Efficiency Graph, and that is an extension of the Ancestral Selection Graph (Krone and Neuhauser (1997a), Krone and Neuhauser (1997b)). The main contribution of this paper is that, in evolving populations, inefficiency can arise as a promoter of selective advantage and not necessarily as a trade-off.

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arXiv2020

Double hypergeometric Lévy processes and self-similarity

Motivated by a recent paper of Budd, where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener-Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

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arXiv2020

Branching processes with interactions: the subcritical cooperative regime

In this paper, we introduce a family of processes with values on the nonnegative integers that describes the dynamics of populations where individuals are allowed to have different types of interactions. The types of interactions that we consider include pairwise interaction, such as competition, annihilation and cooperation; and interaction among several individuals that can be considered as catastrophes. We call such families of processes \textit{branching processes with interactions}. Our aim is to study their long term behaviour under a specific regime of the pairwise interaction parameters that we introduce as {\it subcritical cooperative regime}. Under such regime, we prove that a process in this class comes down from infinity and has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype and whose parameters have a classical interpretation in terms of population genetics. The moment dual is an important tool for characterizing the stationary distribution of branching processes with interactions whenever such distribution exists but it is an interesting object on its own right.

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arXiv2020

Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper, we study the cut-off phenomenon under the total variation distance of $d$-dimensional Ornstein-Uhlenbeck processes which are driven by Lévy processes. That is to say, under the total variation distance, there is an abrupt convergence of the aforementioned process to its equilibrium, i.e. limiting distribution. Despite that the limiting distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases under suitable conditions on the limiting distribution. The cut-off phenomena for the average and superposition processes are also determined.

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arXiv2019

Recurrent extensions of real-valued self-similar Markov processes

Let $X=(X_t, t\geq 0)$ be a self-similar Markov process taking values in $\mathbb{R}$ such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of $X$ that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (2006) and Rivero (2005, 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable Lévy process which to the best of our knowledge has not been studied before.

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arXiv2019

Extinction and coming down from infinity of CB-processes with competition in a Lévy environment

In this note, we are interested on the event of extinction and the property of coming down from infinity of continuous state branching (or CB for short) processes with competition in a Lévy environment whose branching mechanism satisfies the so-called Grey's condition. In particular, we deduce, under the assumption that the Lévy environment does not drift towards infinity, that for any starting point the process becomes extinct in finite time a.s. Moreover if we impose an integrability condition on the competition mechanism, then the process comes down from infinity regardless the long term behaviour of the environment.

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arXiv2019

Extinction time of logistic branching processes in a Brownian environment

In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its own right and provides a one to one correspondence between the latter family of processes and the family of Feller diffusions which are perturbed by an independent spectrally positive Lévy process. When the independent random perturbation (of the Feller diffusion) is driven by a subordinator then the logistic branching processes in a Brownian environment converges to a specified distribution; otherwise, it becomes extinct a.s. In the latter scenario, and following a similar approach to Lambert (Lambert, Ann. Appl. Probab., 2005), we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution to a Ricatti differential equation. In particular, the latter characterises the law of the process coming down from infinity.

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arXiv2018

Backbone decomposition of multitype superprocesses

In this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well-known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.

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arXiv2018

Convergence of the empirical spectral distribution of Gaussian matrix-valued processes

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(λ_{1}^{(n)}(t),\dots, λ_{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $μ^{(n)}=(μ_{t}^{(n)}; t\geq0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $μ^{(n)}$ converges in probability to a deterministic limit $μ$, in the topology of uniform convergence over compact sets. We show that the process $μ$ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers' equation. Our results extend those of Rogers and Shi for the free Brownian motion and Pardo et al. for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.

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arXiv2016

On the Free Fractional Wishart Process

We investigate the process of eigenvalues of a fractional Wishart process defined as N=B*B, where B is a matrix fractional Brownian motion recently studied by Nualart and Pérez-Abreu. Using stochastic calculus with respect to the Young integral we show that the eigenvalues do not collide at any time with probability one. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ we derive a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the free fractional Wishart process which constitutes the family of fractional dilations of the free Poisson distribution.

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arXiv2016

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

Let $ξ=(ξ_t, t\ge 0)$ be a real-valued Lévy process and define its associated exponential functional as follows \[ I_t(ξ):=\int_0^t \exp\{-ξ_s\}{\rm d} s, \qquad t\ge 0. \] Motivated by important applications to stochastic processes in random environment, we study the asymptotic behaviour of \[ \mathbb{E}\Big[F\big(I_t(ξ)\big)\Big] \qquad \textrm{as}\qquad t\to \infty, \] where $F$ is a non-increasing function with polynomial decay at infinity and under some exponential moment conditions on $ξ$. In particular, we find five different regimes that depend on the shape of the Laplace exponent of $ξ$. Our proof relies on a discretisation of the exponential functional $I_t(ξ)$ and is closely related to the behaviour of functionals of semi-direct products of random variables. We apply our main result to three {questions} associated to stochastic processes in random environment. We first consider the asymptotic behaviour of extinction and explosion for {stable} continuous state branching processes in a Lévy random environment. Secondly, we {focus on} the asymptotic behaviour of the mean of a population model with competition in a Lévy random environment and finally, we study the tail behaviour of the maximum of a diffusion process in a Lévy random environment.

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arXiv2016

Continuous state branching processes in random environment: The Brownian case

We consider continuous state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term extinction and explosion behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and, as in the case of branching processes in random environment, we find five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, we study the process conditioned on eventual extinction where three regimes for the asymptotic behaviour of the extinction probability appear. Finally, the process conditioned on non-extinction and the process with immigration are given.

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arXiv2015

Lévy insurance risk processes with parisian type severity of debt

In this article, we introduce a new definition of bankruptcy for a spectrally negative Lévy insurance risk process. More precisely, we study the Gerber-Shiu distribution for a ruin model where at each time the surplus goes negative, an independent negative random level is considered. If a negative excursion of the surplus exceeds such random level then the insurance company goes out of business. Our methodology uses excursion theory and relies on the description of the excursion measure away from 0 which was recently obtained by the authors in Pardo et al. (see arXiv:1507.05225). Our results are given in terms of the so-called scale functions.

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arXiv2015

The excursion measure away from zero for spectrally negative Lévy processes

We provide a description of the excursion measure from a point for a spectrally negative Lévy process. The description is based in two main ingredients. The first is building a spectrally negative Lévy process conditioned to avoid zero and the study of its entrance law at zero. The latter is connected with both the excursion measure from zero of the process reflected in its infimum and reflected in its supremum. This lead us to establish a connection with the excursion measure from the state zero with the excursion measure from zero for the process reflected at the infimum and reflected at the supremum, respectively, which is the second main ingredient of our description.

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arXiv2015

A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion

A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu. Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used.

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arXiv2014

The hitting time of zero for a stable process

For any two-sided jumping $α$-stable process, where $1 < α< 2$, we find an explicit identity for the law of the first hitting time of the origin. This complements existing work in the symmetric case and the spectrally one-sided case; cf. Yano-Yano-Yor (2009) and Cordero (2010), and Peskir (2008) respectively. We appeal to the Lamperti-Kiu representation of Chaumont-Pantí-Rivero (2011) for real-valued self-similar Markov processes. Our main result follows by considering a vector-valued functional equation for the Mellin transform of the integrated exponential Markov additive process in the Lamperti-Kiu representation. We conclude our presentation with some applications.

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arXiv2013

Asymptotic behaviour near extinction of continuous state branching processes

In this note, we study the asymptotic behaviour near extinction of (sub-) critical continuous state branching processes. In particular, we establish an analogue of Khintchin's law of the iterated logarithm near extinction time for a continuous state branching process whose branching mechanism satisfies a given condition and its reflected process at its infimum.

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arXiv2013

Total internal and external lengths of the Bolthausen-Sznitman coalescent

In this paper, we study a weak law of large numbers for the total internal length of the Bolthausen-Szmitman coalescent. As a consequence, we obtain the weak limit law of the centered and rescaled total external length. The latter extends results obtained by Dhersin & Möhle \cite{DM12}. An application to population genetics dealing with the total number of mutations in the genealogical tree is also given.

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