On the asymptotic behaviour of increasing self-similar Markov processes.
Special, conjugate and complete scale functions for spectrally negative Lev́y processes.
Tail asymptotics for exponential functionals of Lévy processes: The convolution equivalent case
We determine the rate of decrease of the right tail distribution of the exponential functional of a Lévy process with a convolution equivalent Lévy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Lévy measure of the underlying Lévy process. The method of proof relies on fluctuation theory of Lévy processes and an explicit pathwise representation of the exponential functional as the exponential functional of a bivariate subordinator....
Sinaıˇ's condition for real valued Lévy processes
Foreword
Real-valued conditional convex risk measures in (ℱ)
The numerical representation of convex risk measures beyond essentially bounded financial positions is an important topic which has been the theme of recent literature. In other direction, it has been discussed the assessment of essentially bounded risks taking explicitly new information into account, i.e., conditional convex risk measures. In this paper we combine these two lines of research. We discuss the numerical representation of conditional...
Bounds of Ruin Probabilities for Insurance Companies in the Presence of Stochastic Volatility on Investments
In this work we consider a model of an insurance company where the insurer has to face a claims process which follows a Compound Poisson process with finite exponential moments. The insurer is allowed to invest in a bank account and in a risky asset described by Geometric Brownian motion with stochastic volatility that depends on an external factor modelled as a diffusion process. By using exponential martingale techniques we obtain upper and lower...
Random Walks and Trees
These notes provide an elementary and self-contained introduction to branching random walks. Section 1 gives a brief overview of Galton–Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-goût to the main part, Section 3, where branching random ...
Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions
We consider the Cauchy problem in ℝ
Fluctuation limit theorems for age-dependent critical binary branching systems
We consider an age-dependent branching particle system in ℝ, where the particles are subject to -stable migration (0 < ≤ 2), critical binary branching, and general (non-arithmetic) lifetimes distribution. The population starts off from a Poisson random field in ℝ with Lebesgue intensity. We prove functional central limit theorems and strong laws of large numbers under two rescalings: high particle density, and a space-time rescaling...
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