In this paper, we study the existence of the transition densities of one-dimensional Lévy processes. Compared with past results, our results contain the Lévy processes whose Lévy symbols have logarithm behavior at infinity. Our results contain the Lévy symbol induced by the following Laplace exponent
$ψ(ξ) := (\ln(1 + \ln(1 + \ln(\cdots \ln(1 + |ξ|)))))^ε$ ($n$ times), $0 < ε< 1$, $2 \le n$.
We also show that $ψ(ξ)$ is a Lévy symbol with transition density.
We prove asymptotic behaviour of transition density for a large class of spectrally one-sided Lévy processes of unbounded variation satisfying mild condition imposed on the second derivative of the Laplace exponent, or equivalently, on the real part of the characteristic exponent. We also provide sharp two-sided estimates on the density when restricted additionally to processes without Gaussian component.
We study a first passage time of a Lévy process over a positive constant level. In the spectrally negative case we give conditions for absolutely continuity of the distributions of the first passage times. The tail asymptotics of their densities are also clarified, where the asymptotics depend on tail behaviour of the corresponding Lévy measures. We apply our results to the mathematical finance, in particular, the credit default swap pricing.
Kuznetsov et al.（2011）和Kuznetsov and Pardo（2013）介绍了超几何Lévy过程族。当人们用正自相似马氏过程理论分析稳定过程时，它们自然地出现在稳定过程波动的研究中。超几何Lévy过程是通过其特征指数来定义的，特征指数是一个复值函数，有四个独立的参数。Kyprianou等人（2014）表明，超几何Lévy过程的定义可以包含比最初指定的更大范围的上述参数。在这篇简短的文章中，我们将参数范围推得更远。特别地，我们计算了Wiener-Hopf因子的潜在Lévy测度和潜在测度。
We find necessary and sufficient conditions for almost sure finiteness of integral functionals of spectrally positive Lévy processes. Via Lamperti type transforms, these results can be applied to obtain new integral tests on extinction and explosion behaviors for a class of continuous-state nonlinear branching processes.
We investigate densities of vaguely continuous convolution semigroups of probability measures on $\mathbb{R}^d$. First, we provide results that give upper estimates in a situation when the corresponding jump measure is allowed to be highly non-symmetric. Further, we prove upper estimates of the density and its derivatives if the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies certain scaling condition. Lower estimates are discussed in view of a recent development in that direction, and in such a way to complement upper estimates. We apply all …
In this short note we study homogenization of symmetric $d$-dimensional Lévy processes. Homogenization of one-dimensional pure jump Markov processes has been investigated by Tanaka \emph{et al.} in 1992; their motivation was the work by Benssousan \emph{et al.}\ from 1975 on the homogenization of diffusion processes in $\mathbb{R}^d$. We investigate a similar problem for a class of symmetric pure-jump Lévy processes on $\mathbb{R}^d$ and we identify -- using Mosco convergence -- the limit process.
We obtain the lower bounds for ergodic convergence rates, including spectral gaps and convergence rates in strong ergodicity for time-changed symmetric Lévy processes by using harmonic function and reversible measure. As direct applications, explicit sufficient conditions for exponential and strong ergodicity are given. Some examples are also presented.
We give new proofs of certain equivalent conditions for the existence of generalized moments of a Lévy process $(X_t)_{t\geq 0}$; in particular, the existence of a generalized $g$-moment is equivalent to the uniform integrability of $(g(X_t))_{t\in [0,1]}$. As a consequence, certain functions of a Lévy process which are integrable and local martingales are already true martingales. Our methods extend to moments of stochastically continuous additive processes, and we give new, short proofs for the characterization of lattice distributions and the transience of Lévy processes.
