arXiv subjects

Victor Rivero

Publications and source records attributed to Victor Rivero.

18 recordsLinked to original sources

arXiv2026

A Tanaka-Type Formula for Compact Sets and Equilibrium Measures of L\'{e}vy Processes

Tanaka's formula is a classical identity for Brownian motion, and Tsukada (2018) extended it to L\'{e}vy processes not necessarily symmetric. From a potential-theoretic point of view, this formula shows that the invariant function for the process killed upon hitting a singleton can be decomposed into the sum of a martingale part and a local time. In this paper, we generalize this singleton setting and derive a Tanaka-type formula for a compact set $B$. To this end, we introduce the equilibrium measure, defined as the rescaled limit of the $q$-capacity measures, and show that the invariant function for the process killed upon hitting $B$ can be represented as the integral, with respect to the equilibrium measure, of the invariant functions associated with processes killed upon hitting singletons, up to an additive constant called the Robin constant. Moreover, when $B$ is an interval, we obtain explicit representations of the equilibrium measure, the Robin constant, and the martingale part for recurrent stable processes as well as for recurrent spectrally negative L\'{e}vy processes. Finally, we discuss how an analogous Tanaka-type formula can also be established for transient L\'{e}vy processes.

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arXiv2025

Norm-dependent Lamperti-type MAP representations of stable processes and Brownian motions in the orthant

We start by remarking a one-to-one correspondence between self-similar Markov processes (ssMps) on a Banach space and Markov additive processes (MAPs) that is analogous to the well-known one between positive ssMps and L\'evy processes through the renowned Lamperti-transform, with the main difference that ours is norm-dependent. We then consider multidimensional self-similar Markov processes obtained by killing or by reflecting a stable process or Brownian motion in the orthant and we then fully describe the MAPs associated to them using the $L_1$-norm. Namely, we describe the MAP underlying the ssMp obtained by killing a $d$-dimensional $\alpha$-stable process when it leaves the orthant and the one obtained by reflecting it back in the orthant continuously (or by a jump); finally, we also describe the MAP underlying $d$-dimensional Brownian motion reflected in the orthant. The first three of the aforementioned examples are pure-jump, and the last is a diffusion, so their characterization is given through their L\'evy system, generator and/or through the modulated SDE that defines them, respectively.

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arXiv2025

The strong law of large numbers and a functional central limit theorem for general Markov additive processes

In this note we re-visit the fundamental question of the strong law of large numbers and central limit theorem for processes in continuous time with conditional stationary and independent increments. For convenience we refer to them as Markov additive processes, or MAPs for short. Historically used in the setting of queuing theory, MAPs have often been written about when the underlying modulating process is an ergodic Markov chain on a finite state space. Recent works have addressed the strong law of large numbers when the underlying modulating process is a general Markov processes. We add to the latter with a different approach based on an ergodic theorem for additive functionals and on the semi-martingale structure of the additive part. This approach also allows us to deal with the setting that the modulator of the MAP is either positive or null recurrent. The methodology additionally inspires a CLT-type result.

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arXiv2024

Stability of (sub)critical non-local spatial branching processes with and without immigration

We consider the setting of either a general non-local branching particle process or a general non-local superprocess, in both cases, with and without immigration. Under the assumption that the mean semigroup has a Perron-Frobenious type behaviour for the immigrated mass, as well as the existence of second moments, we consider necessary and sufficient conditions that ensure limiting distributional stability. More precisely, our first main contribution pertains to proving the asymptotic Kolmogorov survival probability and Yaglom limit for critical non-local branching particle systems and superprocesses under a second moment assumption on the offspring distribution. Our results improve on existing literature by removing the requirement of bounded offspring in the particle setting [21] and generalising [43] to allow for non-local branching mechanisms. Our second main contribution pertains to the stability of both critical and sub-critical non-local branching particle systems and superprocesses with immigration. At criticality, we show that the scaled process converges to a Gamma distribution under a necessary and sufficient integral test. At subcriticality we show stability of the process, also subject to an integral test. In these cases, our results complement classical results for (continuous-time) Galton-Watson processes with immigration and continuous-state branching processes with immigration; see [22,40,42,48,51], among others. In the setting of superprocesses, the only work we know of at this level of generality is summarised in [34]. The proofs of our results, both with and without immigration, appeal to similar technical approaches and accordingly, we include the results together in this paper.

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arXiv2022

Generalized scale functions for spectrally negative Lévy processes

For a spectrally negative Lévy process, scale functions appear in the solution of two-sided exit problems, and in particular in relation with the Laplace transform of the first time it exits a closed interval. In this paper, we consider the Laplace transform of more general functionals, which can depend simultaneously on the values of the process and its supremum up to the exit time. These quantities will be expressed in terms of generalized scale functions, which can be defined using excursion theory. In the case the functional does not depend on the supremum, these scale functions coincide with the ones found on the literature, and therefore the results in this work are an extension of them.

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arXiv2021

Continuity properties and the support of killed exponential functionals

For two independent Lévy processes $ξ$ and $η$ and an exponentially distributed random variable $τ$ with parameter $q>0$, independent of $ξ$ and $η$, the killed exponential functional is given by $V_{q,ξ,η} := \int_0^τ\mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$. Interpreting the case $q=0$ as $τ=\infty$, the random variable $V_{q,ξ,η}$ is a natural generalization of the exponential functional $\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$, the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein-Uhlenbeck process. In this paper we show that also the law of the killed exponential functional $V_{q,ξ,η}$ arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein-Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of $\smash{\int_0^t\mathrm{e}^{-ξ_{s-}}\mathrm{d}η_s}$ for fixed $t\geq0$, as well as for integrals of the form $\smash{\int_0^\infty f(s) \, \mathrm{d}η_s}$ for deterministic functions $f$. Furthermore, applying the same techniques to the case $q=0$, new results on the absolute continuity of the improper integral $\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$ are derived.

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arXiv2020

Stable Lévy processes in a cone

Bañuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) analyse the asymptotic tail distribution of the first time a stable (Lévy) process in dimension $d\geq 2$ exists a cone. We use these results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone). As self-similar Markov processes we examine some of their fundamental properties through the lens of its Lamperti-Kiu decomposition. In particular we are interested to understand the underlying structure of the Markov additive process that drives such processes. As a consequence of our interrogation of the underlying MAP, we are able to provide an answer by example to the open question: If the modulator of a MAP has a stationary distribution, under what conditions does its ascending ladder MAP have a stationary distribution? We show how the two forms of conditioning are dual to one another. Moreover, we construct the recurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of self-similar Markov processes. In the spirit of several very recent works, the results presented here show that many previously unknown results of stable processes, which have long since been understood for Brownian motion, or are easily proved for Brownian motion, become accessible by appealing to the notion of the stable process as a self-similar Markov process, in addition to its special status as a Lévy processes with a semi-tractable potential analysis.

math.PR↗

arXiv2019

Entrance laws at the origin of self-similar Markov processes in high dimensions

In this paper we consider the problem of finding entrance laws at the origin for self-similar Markov processes in $\mathbb{R}^d$, killed upon hitting the origin. Under mild assumptions, we show the existence of an entrance law and the convergence to this law when the process is started close to the origin. We obtain an explicit description of the process started from the origin as the time reversal of the original self-similar Markov process conditioned to hit the origin.

math.PR↗

arXiv2018

Deep factorisation of the stable process III: Radial excursion theory and the point of closest reach

In this paper, we continue our understanding of the stable process from the perspective of the theory of self-similar Markov processes in the spirit of the recent papers of Kyprianou (2016) and Kyprianou et al. (2017). In particular, we turn our attention to the case of $d$-dimensional isotropic stable process, for $d\geq 2$. Using a completely new approach we consider the distribution of the point of closest reach. This leads us to a number of other substantial new results for this class of stable processes. We engage with a new radial excursion theory, never before used, from which we develop the classical Blumenthal-Getoor-Ray identities for first entry/exit into a ball, cf. Blumenthal et al. (1961), to the setting of $n$-tuple laws. We identify explicitly the stationary distribution of the stable process when reflected in its running radial supremum. Moreover, we provide a representation of the Wiener-Hopf factorisation of the MAP that underlies the stable process through the Lamperti-Kiu transform.

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arXiv2017

Deep factorisation of the stable process II; potentials and applications

Here we propose a different perspective of the deep factorisation in Kyprianou (2015) based on determining potentials. Indeed, we factorise the inverse of the MAP-exponent associated to a stable process via the Lamperti-Kiu transform. Here our factorisation is completely independent from the derivation in Kyprianou (2015) , moreover there is no clear way to invert the factors in Kyprianou (2015) to derive our results. Our method gives direct access to the potential densities of the ascending and descending ladder MAP of the Lamperti-stable MAP in closed form. In the spirit of the interplay between the classical Wiener-Hopf factorisation and fluctuation theory of the underlying Levy process, our analysis will produce a collection of of new results for stable processes. We give an identity for the point of closest reach to the origin for a stable process with index $α\in (0,1)$ as well as and identity for the point of furthest reach before absorption at the origin for a stable process with index $α\in (1,2)$. Moreover, we show how the deep factorisation allows us to compute explicitly the stationary distribution of stable processes multiplicatively reflected in such a way that it remains in the strip [-1,1].

math.PR↗

arXiv2016

Conditioning subordinators embedded in Markov processes

The running infimum of a Levy process relative to its point of issue is know to have the same range that of the negative of a certain subordinator. Conditioning a Levy process issued from a strictly positive value to stay positive may therefore be seen as implicitly conditioning its descending ladder heigh subordinator to remain in a strip. Motivated by this observation, we consider the general problem of conditioning a subordinator to remain in a strip. Thereafter we consider more general contexts in which subordinators embedded in the path decompositions of Markov processes are conditioned to remain in a strip.

math.PR↗

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.

math.PR↗

arXiv2014

Asymptotic behaviour of first passage time distributions for subordinators

In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt in 1987. In the particular case of a subordinator in the domain of attraction of a stable distribution the results are the analogue of the results obtained by the authors for non-monotone Lévy processes. For subordinators an approach different to that used for non-monotone Lévy processes is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential.

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arXiv2011

Asymptotic behaviour of first passage time distributions for Lévy processes

Let $X$ be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function $c.$ As an analogue of the random walk results in \cite{vw} and \cite{rad} we study the local behaviour of the distribution of the lifetime $ζ$ under the characteristic measure $\underline{n}$ of excursions away from 0 of the process $X$ reflected in its past infimum, and of the first passage time of $X$ below $0,$ $T_{0}=\inf \{t>0:X_{t}<0\},$ under $\mathbb{P}_{x}(\cdot),$ for $x>0,$ in two different regimes for $x,$ viz. $x=o(c(\cdot))$ and $x>D c(\cdot),$ for some $D>0.$ We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at $T_{0}$ and discontinuous passage. In the way to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to $X$ reflected in its past infimum.

math.PR↗

arXiv2011

On the density of exponential functionals of Lévy processes

In this paper, we study the existence of the density associated to the exponential functional of the Lévy process $ξ$, \[ I_{\ee_q}:=\int_0^{\ee_q} e^{ξ_s} \, \mathrm{d}s, \] where $\ee_q$ is an independent exponential r.v. with parameter $q\geq 0$. In the case when $ξ$ is the negative of a subordinator, we prove that the density of $I_{\ee_q}$, here denoted by $k$, satisfies an integral equation that generalizes the one found by Carmona et al. \cite{Carmona97}. Finally when $q=0$, we describe explicitly the asymptotic behaviour at 0 of the density $k$ when $ξ$ is the negative of a subordinator and at $\infty$ when $ξ$ is a spectrally positive Lévy process that drifts to $+\infty$.

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arXiv2011

The theory of scale functions for spectrally negative Le vy processes

The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Levy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Levy processes, in particular a reasonable understanding of the Levy-Khintchine formula and its relationship to the Levy-Ito decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Levy processes; Bertoin (1996), Sato (1999), Applebaum (2004), Kyprianou (2006) and Doney (2007).

math.PR↗

arXiv2007

On the asymptotic behaviour of increasing positive self-similar Markov processes

We are interested by the rate of growth of increasing positive self-similar Markov processes (ipssMp) such that the subordinator associated to it via Lamperti's transformation has infinite mean. We prove that the logarithm of an ipssMp normalized by the logarithm of the time converges weakly, as the time tends to infinity, if and only if the Laplace exponent of the underlying subordinator is regularly varying at zero. Moreover, we prove that the regular variation at zero of the Laplace exponent is essentially nasc for the existence of a function that normalizes the logarithm of an ipssMp. We obtain a law of iterated logarithm for the liminf of the logarithm of an ipssMp and an integral test to study the upper envelope of it. Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained.

math.PR↗

arXiv2005

Sinai's condition for real valued Lévy processes

We prove that the upward ladder height subordinator $H$ associated to a real valued Lévy process $ξ$ has Laplace exponent $ϕ$ that varies regularly at $\infty$ (resp. at 0) if and only if the underlying Lévy process $ξ$ satisfies Sinai's condition at 0 (resp. at $\infty$). Sinai's condition for real valued Lévy processes is the continuous time analogue of Sinai's condition for random walks. We provide several criteria in terms of the characteristics of $ξ$ to determine whether or not it satisfies Sinai's condition. Some of these criteria are deduced from tail estimates of the Lévy measure of $H,$ here obtained, and which are analogous to the estimates of the tail distribution of the ladder height random variable of a random walk which are due to Veraverbeke and Grübel

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