arXiv subjects

Andreas E. Kyprianou

Publications and source records attributed to Andreas E. Kyprianou.

At least 55 records · Page 3Linked to original sources

arXiv2015

Deep factorisation of the stable process

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 $α$-stable process with $α\in(0,2)$, 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 Lévy 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.

math.PR↗

arXiv2015

The Seneta-Heyde scaling for homogeneous fragmentations

Homogeneous mass fragmentation processes describe the evolution of a unit mass that breaks down randomly into pieces as time. Mathematically speaking, they can be thought of as continuous-time analogues of branching random walks with non-negative displacements. Following recent developments in the theory of branching random walks, in particular the work of \cite{AShi10}, we consider the problem of the Seneta-Heyde norming of the so-called additive martingale at criticality. Aside from replicating results for branching random walks in the new setting of fragmentation processes, our main goal is to present a style of reasoning, based on $L^p$ estimates, which works for a whole host of different branching-type processes. We show that our methods apply equally to the setting of branching random walks, branching Brownian motion as well as Gaussian multiplicative chaos.

math.PR↗

arXiv2015

Perpetual Integrals for Levy Processes

We ask for necessary and sufficient conditions for almost sure finiteness of the perpetual integrals of a Levy process. Zero-one laws are already known for Brownian motion with drift and spectrally one-sided Levy processes. Under the assumption that local times exist, we use fluctuation theory and Jeulin's lemma to prove the zero-one law.

math.PR↗

arXiv2015

Real Self-Similar Processes Started from the Origin

Since the seminal work of Lamperti there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-similar Markov processes (rssMps) there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti-Kiu representation to the origin.

math.PR↗

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.

math.PR↗

arXiv2013

Spines, skeletons and the Strong Law of Large Numbers for superdiffusions

Consider a supercritical superdiffusion (X_t) on a domain D subset R^d with branching mechanism -β(x) z+α(x) z^2 + int_{(0,infty)} (e^{-yz}-1+yz) Pi(x,dy). The skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion. We use this decomposition to derive the Strong Law of Large Numbers (SLLN) for a wide class of superdiffusions from the corresponding result for branching particle diffusions. That is, we show that for suitable test functions f and starting measures mu, < f,X_t>/P_{mu}[< f,X_t>] -> W_{infty}, P_{mu}-almost surely as t->infty, where W_{infty} is a finite, non-deterministic random variable characterised as a martingale limit. Our method is based on skeleton and spine techniques and offers structural insights into the driving force behind the SLLN for superdiffusions. The result covers many of the key examples of interest and, in particular, proves a conjecture by Fleischmann and Swart for the super-Wright-Fisher diffusion.

math.PR↗

arXiv2013

The total mass of super-Brownian motion upon exiting balls and Sheu's compact support condition

We study the total mass of a d-dimensional super-Brownian motion as it first exits an increasing sequence of balls. The process of the total mass is a time-inhomogeneous continuous-state branching process, where the increasing radii of the balls are taken as the time parameter. We are able to characterise its time-dependent branching mechanism and show that it converges, as time goes to infinity, towards the branching mechanism of the total mass of a one-dimensional super-Brownian motion as it first crosses above an increasing sequence of levels. Our results allow us to identify the compact support criterion given in Sheu (1994) as a classical Grey condition (1974) for the aforementioned limiting branching mechanism.

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arXiv2013

Multilevel Monte Carlo simulation for Levy processes based on the Wiener-Hopf factorisation

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of Levy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we provide here for the first time a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant for computing expectations of functions depending on the historical trajectory of a Levy process. We derive rates of convergence for both methods and show that they are uniform with respect to the "jump activity" (e.g. characterised by the Blumenthal-Getoor index). We also present a modified version of the algorithm in Kuznetsov et al. (2011) which combined with the multilevel methodology obtains the optimal rate of convergence for general Levy processes and Lipschitz functionals. This final result is only a theoretical one at present, since it requires independent sampling from a triple of distributions which is currently only possible for a limited number of processes.

math.PR↗

arXiv2012

Branching Brownian motion in strip: survival near criticality

We consider a branching Brownian motion with linear drift in which particles are killed on exiting the interval (0,K) and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible. We combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near-critical survival probability. This allows us to deduce the existence of a quasi-stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality. This paper is motivated by recent work on survival of near critical branching Brownian motion with absorption at the origin by Aidekon and Harris as well as the work of Berestycki, Berestycki and Schweinsberg.

math.PR↗

arXiv2012

Survival of homogeneous fragmentation processes with killing

We consider a homogenous fragmentation process with killing at an exponential barrier. With the help of two families of martingales we analyse the growth of the largest fragment for parameter values that allow for survival. In this respect the present paper is also concerned with the probability of extinction of the killed process.

math.PR↗

arXiv2012

Hitting distributions of alpha-stable processes via path censoring and self-similarity

In this paper we return to the problem of Blumenthal-Getoor-Ray, published in 1961, which gave the law of the position of first entry of a symmetric alpha-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional alpha-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between alpha-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of a type of path censoring. Second, we are able to describe in explicit analytical detail a non-trivial Wiener-Hopf factorisation of an auxiliary Levy process from which the desired solution can be sourced. Moreover, as a consequence of this approach, we are able to deliver a number of additional, related identities in explicit form for alpha-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 capped optimal stopping problem for the maximum process

This paper concerns an optimal stopping problem driven by the running maximum of a spectrally negative Levy process X. More precisely, we are interested in capped versions of the American lookback optimal stopping problem, which has its origins in mathematical finance, and provide semi-explicit solutions in terms of scale functions. The optimal stopping boundary is characterised by an ordinary first-order differential equation involving scale functions and, in particular, changes according to the path variation of X. Furthermore, we will link these capped problems to Peskir's maximality principle.

math.PR↗

arXiv2012

Spectrally negative Levy processes perturbed by functionals of their running supremum

In the setting of the classical Cramer-Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if $X = \{X_t : t\geq 0\}$ represents the Cramer-Lundberg process and, for all $t\geq 0$, $S_t = \sup_{s\leq t}X_s$, then Albrecher and Hipp (2007) study $X_t - γS_t$, $t\geq 0$, where $γ\in(0,1)$ is the rate at which tax is paid. This model has been generalised to the setting that $X$ is a spectrally negative Lévy process by Albrecher et al. \cite{albr_ren_zhou}. Finally Kyprianou and Zhou (2009) extend this model further by allowing the rate at which tax is paid with respect to the process $S = \{S_t : t\geq 0\}$ to vary as a function of the current value of $S$. Specifically, they consider the so-called perturbed spectrally negative Levy process, \[ U_t=X_t-\int_{(0,t]}γ(S_u)\,{\rm d} S_u,\qquad t\geq 0, \] under the assumptions $γ:[0,\infty)\rightarrow [0,1)$ and $\int_0^\infty (1-γ(s)){\rm d}s =\infty$. In this article we show that a number of the identities in Kyprianou and Zhou (2009) are still valid for a much more general class of rate functions $γ:[0,\infty)\rightarrow \mathbb{R}$. Moreover, we show that, with appropriately chosen $γ$, the perturbed process can pass continuously (ie. creep) into $(-\infty, 0)$ in two different ways.

math.PR↗

arXiv2011

Meromorphic Levy processes and their fluctuation identities

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener-Hopf factorization for Levy processes where previously there had been very few. We mention in particular the many cases of spectrally negative Levy processes, hyper-exponential and generalized hyper-exponential Levy processes, Lamperti-stable processes, Hypergeometric processes, Beta-processes and Theta-processes. In this paper we introduce a new family of Levy processes, which we call Meromorphic Levy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of their Wiener-Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Levy-Khintchin exponent, all of which appear on the imaginary axis of the complex plane. The specific structure of the M-class Wiener-Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance, insurance risk theory and queuing theory.

math.PR↗

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↗

arXiv2011

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein-Uhlenbeck process associated with Bertoin's tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Levy processes.

math.PR↗

arXiv2010

A transformation for Lévy processes with one-sided jumps and applications

The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts by Patie, Kyprianou and Patie, and, Gnedin, as well as in older work of Urbanik . We show how some specific instances of this mapping prove to be useful for a variety of applications.

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