arXiv subjects

Andreas E. Kyprianou

Publications and source records attributed to Andreas E. Kyprianou.

62 records · Page 4Linked to original sources

arXiv2010

The prolific backbone for supercritical superdiffusions

We develop an idea of Evans and O'Connell, Englander and Pinsky and Duquesne and Winkel by giving a pathwise construction of the so called `backbone' decomposition for supercritical superprocesses. Our results also complement a related result for critical $(1+β)$-superprocesses given in Etheridge and Williams \cite{EW}. Our approach takes an analytical point of view which is more in the spirit of the original Evans and O'Connell paper.

math.PR↗

arXiv2010

Optimal control with absolutely continuous strategies for spectrally negative Levy processes

In the last few years there has been renewed interest in the classical control problem of de Finetti for the case that underlying source of randomness is a spectrally negative Levy process. In particular a significant step forward is made in an article of Loeffen where it is shown that a natural and very general condition on the underlying Levy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Levy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem but now with the restriction that control strategies are absolutely continuous with respect to Lebesgue measure. This problem has been considered by Asmussen and Taksar, Jeanblanc and Shiryaev and Boguslavskaya in the diffusive case and Gerber and Shiu for the case of a Cramer-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Levy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.

math.PR↗

arXiv2010

On the excursions of reflected local time processes and stochastic fluid queues

This paper extends previous work by the authors. We consider the local time process of a strong Markov process, add negative drift, and reflect it à la Skorokhod. The resulting process is used to model a fluid queue. We derive an expression for the joint law of the duration of an excursion, the maximum value of the process on it, and the time distance between successive excursions. We work with a properly constructed stationary version of the process. Examples are also given in the paper.

math.PR↗

arXiv2009

General tax structures and the Levy insurance risk model

In the spirit of previous of Albrecher, Hipp, Renaud and Zhou we consider a Lévy insurance risk model with tax payments of a more general structure than in the aforementioned papers that was also considered in \cite{ABBR}. In terms of scale functions, we establish three fundamental identities of interest which have stimulated a large volume of actuarial research in recent years. That is to say, the two sided exit problem, the net present value of tax paid until ruin as well as a generalized version of the Gerber-Shiu function. The method we appeal to differs from former works in that we appeal predominantly to excursion theory.

math.PR↗

arXiv2008

Refracted Levy processes

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter 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 formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation \[ \D U_t = - δ\mathbf{1}_{\{U_t >b\}}\D t + \D X_t \] where $X=\{X_t :t\geq 0\}$ is a Lévy process with law $\mathbb{P}$ and $b, δ\in \mathbb{R}$ such that the resulting process $U$ may visit the half line $(b,\infty)$ with positive probability. We consider in particular the case that $X$ is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the $q$-scale function of the driving Lévy process and its perturbed version describing motion above the level $b$. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

math.PR↗

arXiv2007

Special, conjugate and complete scale functions for spectrally negative Lévy processes

Following from recent developments by Hubalek and Kyprianou, the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative Lévy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative Lévy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed.

math.PR↗

arXiv2007

Strong Law of Large Numbers for branching diffusions

Let $X$ be the branching particle diffusion corresponding to the operator $Lu+β(u^{2}-u)$ on $D\subseteq \mathbb{R}^{d}$ (where $β\geq 0$ and $β\not\equiv 0$). Let $λ_{c}$ denote the generalized principal eigenvalue for the operator $L+β$ on $D$ and assume that it is finite. When $λ_{c}>0$ and $L+β-λ_{c}$ satisfies certain spectral theoretical conditions, we prove that the random measure $\exp \{-λ_{c}t\}X_{t}$ converges almost surely in the vague topology as $t$ tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of \cite{ET,EW}. We extend significantly the results in \cite{AH76,AH77} and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and `spine' decompositions or `immortal particle pictures'.

math.PR↗

arXiv2005

Ruin Probabilities and Overshoots for General Levy Insurance Risk Processes

We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to -\infty a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Levy processes.

math.PR↗