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....
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...
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...
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
...
We consider the Cauchy problem in ℝ
d
for a class of
semilinear parabolic partial differential equations that arises in some stochastic control
problems. We assume that the coefficients are unbounded and locally Lipschitz, not
necessarily differentiable, with continuous data and local uniform ellipticity. We
construct a classical solution by approximation with linear parabolic equations....
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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