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J. C. Pardo

Publications and source records attributed to J. C. Pardo.

16 recordsLinked to original sources
arXiv2025

Cutoff stability of multivariate geometric Brownian motion

This article establishes cutoff convergence or abrupt convergence of three statistical quantities for multivariate (Hurwitz) stable geometric Brownian motion: the autocorrelation function, the Wasserstein distance between the current state and its degenerate limiting measure, and, finally, anti-concentration probabilities, which yield a fine-tuned trade-off between almost sure rates and the respective integrability of the random modulus of convergence using a quantitative Borel--Cantelli Lemma. We obtain in case of simultaneous diagonalizable drift and volatility matrices a complete representation of the mean square and derive nontrivial, sufficient and necessary mean square stability conditions, which include all real and imaginary parts of the volatility matrices' spectra.

math.PR↗
arXiv2022

The cutoff phenomenon for the stochastic heat and the wave equation subject to small Lévy noise

This article generalizes the small noise cutoff phenomenon to the strong solutions of the stochastic heat equation and the damped stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and Lévy noises in the Wasserstein distance. For the additive noise case, we obtain analogous infinite dimensional results to the respective finite dimensional cases obtained recently by Barrera, Högele and Pardo (JSP2021), that is, the (stronger) profile cutoff phenomenon for the stochastic heat equation and the (weaker) window cutoff phenomenon for the stochastic wave equation. For the multiplicative noise case, which is studied in this context for the first time, the stochastic heat equation also exhibits profile cutoff phenomenon, while for the stochastic wave equation the methods break down due to the lack of symmetry. The methods rely strongly on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the stochastic solution flows in terms of stochastic exponentials.

math.PR↗
arXiv2021

The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise

This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump Lévy noise of small amplitude $\varepsilon>0$, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes $X^\varepsilon$ turns out to be exponentially ergodic with equilibrium distribution $μ^{\varepsilon}$ in total variation distance which extends a result from Peng and Zhang (2018) to arbitrary polynomial moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index $α>3/2$. That is to say, in this setting we identify a deterministic time scale $\mathfrak{t}_{\varepsilon}^{\mathrm{cut}}$ satisfying $\mathfrak{t}_ \varepsilon^{\mathrm{cut}} \rightarrow \infty$, as $\varepsilon \rightarrow 0$, and a respective time window, $\mathfrak{t}_\varepsilon^{\mathrm{cut}} \pm o(\mathfrak{t}_\varepsilon^{\mathrm{cut}})$, during which the total variation distance between the current state and its equilibrium $μ^{\varepsilon}$ essentially collapses as $\varepsilon$ tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in Barrera and Jara (2020) to the Lévy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric $α$-stable noise for $α>3/2$. The proof techniques differ completely from the Gaussian case due to the absence of respective Girsanov transforms which couple the nonlinear equation and the linear approximation asymptotically even for short times.

math.PR↗
arXiv2020

Distributional properties of exponential functionals of Levy processes

We study the distribution of the exponential functional $I(ξ,η)=\int_0^{\infty} \exp(ξ_{t-}) \d η_t$, where $ξ$ and $η$ are independent Lévy processes. In the general setting using the theories of Markov processes and Schwartz distributions we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in Carmona et al "On the distribution and asymptotic results for exponential functionals of Levy processes". In the special case when $η$ is a Brownian motion with drift we show that this integral equation leads to an important functional equation for the Mellin transform of $I(ξ,η)$, which proves to be a very useful tool for studying the distributional properties of this random variable. For general Lévy process $ξ$ ($η$ being Brownian motion with drift) we prove that the exponential functional has a smooth density on $\r \setminus \{0\}$, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption that $ξ$ has some positive exponential moments we establish an asymptotic behaviour of $\p(I(ξ,η)>x)$ as $x\to +\infty$, and under similar assumptions on the negative exponential moments of $ξ$ we obtain a precise asympotic expansion of the density of $I(ξ,η)$ as $x\to 0$. Under further assumptions on the Lévy process $ξ$ one is able to prove much stronger results about the density of the exponential functional and we illustrate some of the ideas and techniques for the case when $ξ$ has hyper-exponential jumps.

math.PR↗
arXiv2016

Branching processes in a Lévy random environment

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by Brownian motions and Poisson random measures which are mutually independent. The existence and uniqueness of strong solutions are established under some general conditions that allows us to consider the case when the strong solution explodes at a finite time. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study some properties of such processes that extends recent results obtained by Bansaye et al. in (Electron. J. Probab. 18, no. 106, 1-31, (2013)), Palau and Pardo in (arXiv:1506.09197 (2015)) and Evans et al. in (J. Math. Biol., 71, 325-359, (2015)).

math.PR↗
arXiv2015

Gerber-Shiu functionals at Parisian ruin for Lévy insurance risk processes

Inspired by works of Landriault et al. \cite{LRZ-0, LRZ}, we study discounted penalties at ruin for surplus dynamics driven by a spectrally negative Lévy process with Parisian implementation delays. To be specific, we study the so-called Gerber-Shiu functional for a ruin model where at each time the surplus process goes negative, an independent exponential clock with rate $q>0$ is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of the so-called scale functions. In particular, our results extend recent results of Landriault et al. \cite{LRZ-0, LRZ}.

math.PR↗
arXiv2014

The extended hypergeometric class of Lévy processes

With a view to computing fluctuation identities related to stable processes, we review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (arXiv:1012.0817). We give the Wiener-Hopf factorisation of a process in the extended class, and characterise its exponential functional. Finally, we give three concrete examples arising from transformations of stable processes.

math.PR↗
arXiv2012

Occupation times of refracted Lévy processes

A refracted Lévy process is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation \[ \ud U_t=-δ\mathbf{1}_{\{U_t>b\}}\ud t +\ud X_t, \] where $X=(X_t, t\ge 0)$ is a Lévy process with law $\p$ and $b,δ\in \R$ such that the resulting process $U$ may visit the half line $(b,\infty)$ with positive probability. In this paper, we consider the case that $X$ is spectrally negative and establish a number of identities for the following functionals \[ \int_0^\infty\mathbf{1}_{\{U_t a\}$ and $ρ^-_c=\inf\{t\ge 0: U_t< c\}$ for $c<b<a$. Our identities extend recent results of Landriault et al. \cite{LRZ} and bear relevance to Parisian-type financial instruments and insurance scenarios.

math.PR↗
arXiv2012

A Wiener--Hopf Monte Carlo simulation technique for Lévy processes

We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr's so-called "Canadization" technique as well as Doney's method of stochastic bounds for Lévy processes; see Carr [Rev. Fin. Studies 11 (1998) 597--626] and Doney [Ann. Probab. 32 (2004) 1545-1552]. We rely fundamentally on the Wiener-Hopf decomposition for Lévy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos Lévy en titillant la factorization de Wiener-Hopf (2002) Laboratoire de Mathématiques de L'INSA de Rouen] and Kuznetsov [Ann. Appl. Probab. 20 (2010) 1801--1830]. We illustrate our Wiener--Hopf Monte Carlo method on a number of different processes, including a new family of Lévy processes called hypergeometric Lévy processes. Moreover, we illustrate the robustness of working with a Wiener--Hopf decomposition with two extensions. The first extension shows that if one can successfully simulate for a given Lévy processes then one can successfully simulate for any independent sum of the latter process and a compound Poisson process. The second extension illustrates how one may produce a straightforward approximation for simulating the two-sided exit problem.

math.PR↗
arXiv2009

Explicit identities for Lévy processes associated to symmetric stable processes

In this paper we introduce a new class of Lévy processes which we call hypergeometric-stable Lévy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Lévy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.

math.PR↗
arXiv2009

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

Understanding the space-time features of how a Lévy 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 Lévy 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 Lévy processes, Lévy 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 Lévy 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) Lévy processes and a positive self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable Lévy processes. This leads further to the introduction of a more general family of Lévy processes which we call hypergeometric Lévy processes, for which similar explicit identities may be considered.

math.PR↗
arXiv2009

The Gapeev-Kühn stochastic game driven by a spectrally positive Lévy process

In Gapeev and Kühn (2005), the stochastic game corresponding to perpetual convertible bonds was considered when driven by a Brownian motion and a compound Poisson process with exponential jumps. We consider the same stochastic game but driven by a spectrally positive Lévy process. We establish a complete solution to the game indicating four principle parameter regimes as well as characterizing the occurence of continuous and smooth fit. In Gapeev and Kühn (2005), the method of proof was mainly based on solving a free boundary value problem. In this paper, we instead use fluctuation theory and an auxiliary optimal stopping problem to find a solution to the game.

math.PR↗
arXiv2008

On the Lamperti stable processes

We consider a new family of $\R^d$-valued Lévy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance \cite{cc}, \cite{ckp}, \cite{kp} and \cite{pp}). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosiński \cite{ro} and Houdré and Kawai \cite{hok} respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simulation. In this work we find general properties of this class and we also provide many examples, some of which appear in recent literature.

math.PR↗
arXiv2007

On continuous state branching processes: conditioning and self-similarity

In this paper, for $α\in (1, 2}$ we show that the $α$-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.

math.PR↗
arXiv2006

On the future infimum of positive self-similar Markov processes

We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and infinity. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont and Pardo [9]. These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan et al. [11].

math.PR↗