Results 1  10
of
38
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
Abstract

Cited by 84 (17 self)
 Add to MetaCart
Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
Abstract

Cited by 48 (3 self)
 Add to MetaCart
Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Symmetric jump processes and their heat kernel estimates
 Sci. China Ser. A
"... We survey the recent development of the DeGiorgiNashMoserAronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integrodifferential operators). We focus on the sharp twosided estimates for the transition density functions (or heat kernels) of the proc ..."
Abstract

Cited by 18 (9 self)
 Add to MetaCart
(Show Context)
We survey the recent development of the DeGiorgiNashMoserAronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integrodifferential operators). We focus on the sharp twosided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integrodifferential operators are mainly probabilistic.
A transformation from Hausdorff to Stieltjes moment sequences
 Ark. Mat
, 2004
"... Abstract We introduce a nonlinear injective transformation T from the set of nonvanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula T [(an)]n = 1/(a1 *... * an). Special cases of this transformation have appeared in various papers on e ..."
Abstract

Cited by 13 (6 self)
 Add to MetaCart
(Show Context)
Abstract We introduce a nonlinear injective transformation T from the set of nonvanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula T [(an)]n = 1/(a1 *... * an). Special cases of this transformation have appeared in various papers on exponential functionals of L'evy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to qseries. 2000 Mathematics Subject Classification: primary 44A60; secondary 33D65. Keywords: moment sequence, qseries. 1 Introduction and main results In his fundamental memoir [23] Stieltjes characterized sequences of the form sn = Z 1
Ruin probabilities for L'evy processes with mixedexponential negative jumps
 Theory of Probability and its Applications
, 1999
"... Closed form of the ruin probability for a L'evy processes, possible killed at a constant rate, with arbitrary positive, and mixed exponentially negative jumps is given. Keywords: Ruin probability, closed form, L'evy process, mixedexponential distributions. 1 Introduction 1.1 Let X = fX ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
Closed form of the ruin probability for a L'evy processes, possible killed at a constant rate, with arbitrary positive, and mixed exponentially negative jumps is given. Keywords: Ruin probability, closed form, L'evy process, mixedexponential distributions. 1 Introduction 1.1 Let X = fX t g t0 be a real valued stochastic process defined on a stochastic basis(\Omega ; F ; F = (F t ) t0 ; P ) that satisfy the usual conditions. Assume that X is c`adl`ag, adapted, X 0 = 0, and for 0 s ! t the random variable X t \Gamma X s is independent of the oefield F s with a distribution that only depends on the difference t \Gamma s. X is a process with stationary independent increments (PIIS), or a L'evy process. For q 2 R, /(q) denotes the characteristic exponent of X given by L'evyKhinchine formula /(q) = 1 t log E(e iqX t ) = ibq \Gamma 1 2 oe 2 q 2 + Z R (e iqy \Gamma 1 \Gamma iqy1 fjyj!1g )\Pi(dy) where b and oe 0 are real constants, and \Pi is a positive measure on R \...
Towards Practical and Synthetical Modelling of Repairable Systems
"... Abstract: In this article, we survey the developments with respect to generalized models of repairable systems during the 1990s, particularly for the last five years. In this field, we notice the sharp fundamental problem that voluminous and complicated models are proposed without sufficient evidenc ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
Abstract: In this article, we survey the developments with respect to generalized models of repairable systems during the 1990s, particularly for the last five years. In this field, we notice the sharp fundamental problem that voluminous and complicated models are proposed without sufficient evidence (or data) for justifying a success in tackling real engineering problems. Instead of following the myth of using simple models to face complicated reality, and based on our own research experiences, we select and review some practical models, in the quickly growing areas: age models, condition monitoring models, and shock and wear models, including the delaytime models. Further, we also notice that there is an attempt to develop synthetical models from a different point of view. Therefore, we comment the relevant developments with strong emphasis on stochastic processes reflecting the intrinsic nature of the actual physical dynamics of those repairable system models. 1
On the rerooting invariance property of Lévy trees
, 2009
"... We prove a strong form of the invariance under rerooting of the distribution of the continuous random trees called Lévy trees. This extends previous results due to several authors. 1 ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
We prove a strong form of the invariance under rerooting of the distribution of the continuous random trees called Lévy trees. This extends previous results due to several authors. 1
PutCall Duality and Symmetry
, 2003
"... The aim of this work is to examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov’s Theorem, is called put–call duality. It in ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The aim of this work is to examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov’s Theorem, is called put–call duality. It includes as a particular case, the relation known as put–call symmetry. Necessary and sufficient conditions for put–call symmetry to hold are obtained, in terms of the triplet of predictable characteristic of the Lévy process, answering in this way a question raised by Carr and Chesney (1996).
Bayesian Analysis of Stochastic Volatility Models with Levy Jumps: Application to Value at Risk
, 2007
"... In this paper we analyze asset returns models with diffusion part and jumps in returns with stochastic volatility either from diffusion or pure jump part. We consider different specifications for the pure jump part including compound Poisson, Variance Gamma and Levy αstable jumps. Monte Carlo Marko ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
In this paper we analyze asset returns models with diffusion part and jumps in returns with stochastic volatility either from diffusion or pure jump part. We consider different specifications for the pure jump part including compound Poisson, Variance Gamma and Levy αstable jumps. Monte Carlo Markov chain algorithm is constructed to estimate models with latent Variance Gamma and Levy α−stable jumps. Our construction corrects for separability problems in the state space of the MCMC for Levy α−stable jumps. We apply our model specifications for analysis of S&P500 daily returns. We find, that models with infinite activity jumps and stochastic volatility from diffusion perform well in capturing S&P500 returns characteristics. Models with stochastic volatility from jumps cannot represent excess kurtosis and tails of returns distributions. Oneday and oneweek ahead prediction and VaR performance characterizing conditional returns distribution rejects Variance Gamma jumps in favor of Levy α−stable jumps in returns. JEL classification: C1; C11; G1; G12 1 1