scholarly journals On Maxima and Ladder Processes for a Dense Class of Lévy Process

2006 ◽  
Vol 43 (1) ◽  
pp. 208-220 ◽  
Author(s):  
Martijn Pistorius
Keyword(s):  
Laplace Transform ◽  
Error Estimates ◽  
Lévy Process ◽  
Iterative Procedure ◽  
Levy Process ◽  
Distribution Of The Maximum ◽  
Phase Type ◽  
The Laplace Transform ◽  
The Law ◽  
Dense Class

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

scholarly journals On Maxima and Ladder Processes for a Dense Class of Lévy Process

2006 ◽  
Vol 43 (01) ◽  
pp. 208-220 ◽  
Author(s):  
Martijn Pistorius
Keyword(s):  
Laplace Transform ◽  
Error Estimates ◽  
Lévy Process ◽  
Iterative Procedure ◽  
Levy Process ◽  
Distribution Of The Maximum ◽  
Phase Type ◽  
The Laplace Transform ◽  
The Law ◽  
Dense Class

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

scholarly journals Last Exit Before an Exponential Time for Spectrally Negative Lévy Processes

2009 ◽  
Vol 46 (02) ◽  
pp. 542-558 ◽  
Author(s):  
E. J. Baurdoux
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Risk Theory ◽  
Exponential Time ◽  
Main Motivation ◽  
Spectrally Negative Lévy Process ◽  
The Laplace Transform ◽  
Spectrally Negative Lévy Processes

Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).

scholarly journals Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view

2016 ◽  
Vol 48 (1) ◽  
pp. 274-297 ◽  
Author(s):  
Hélène Guérin ◽  
Jean-François Renaud
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Joint Distribution ◽  
Jump Diffusion ◽  
Levy Process ◽  
Underlying Process ◽  
Scale Functions ◽  
Spectrally Negative Lévy Process ◽  
The Laplace Transform ◽  
Jump Diffusion Model

Abstract We study the distribution Ex[exp(-q∫0t1(a,b)(Xs)ds); Xt ∈ dy], where -∞ ≤ a < b < ∞, and where q, t > 0 and x ∈ R for a spectrally negative Lévy process X. More precisely, we identify the Laplace transform with respect to t of this measure in terms of the scale functions of the underlying process. Our results are then used to price step options and the particular case of an exponential spectrally negative Lévy jump-diffusion model is discussed.

Some fluctuation identities for Lévy processes with jumps of the same sign

2004 ◽  
Vol 41 (04) ◽  
pp. 1191-1198 ◽  
Author(s):  
Xiaowen Zhou
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Joint Distribution ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Exit Problem ◽  
The Laplace Transform ◽  
The One

We consider a two-sided exit problem for a Lévy process with no positive jumps. The Laplace transform of the time when the process first exits an interval from above is obtained. It is expressed in terms of another Laplace transform for the one-sided exit problem. Applications of this result are discussed. In particular, a new expression for the solution to the two-sided exit problem is obtained. The joint distribution of the minimum and the maximum values of such a Lévy process is also studied.

scholarly journals Last Exit Before an Exponential Time for Spectrally Negative Lévy Processes

2009 ◽  
Vol 46 (2) ◽  
pp. 542-558 ◽  
Author(s):  
E. J. Baurdoux
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Risk Theory ◽  
Exponential Time ◽  
Main Motivation ◽  
Spectrally Negative Lévy Process ◽  
The Laplace Transform ◽  
Spectrally Negative Lévy Processes

Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).

Some fluctuation identities for Lévy processes with jumps of the same sign

2004 ◽  
Vol 41 (4) ◽  
pp. 1191-1198 ◽  
Author(s):  
Xiaowen Zhou
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Joint Distribution ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Exit Problem ◽  
The Laplace Transform ◽  
The One

We consider a two-sided exit problem for a Lévy process with no positive jumps. The Laplace transform of the time when the process first exits an interval from above is obtained. It is expressed in terms of another Laplace transform for the one-sided exit problem. Applications of this result are discussed. In particular, a new expression for the solution to the two-sided exit problem is obtained. The joint distribution of the minimum and the maximum values of such a Lévy process is also studied.

scholarly journals A factorization of a Lévy process over a phase-type horizon

2018 ◽  
Vol 34 (4) ◽  
pp. 397-408 ◽  
Author(s):  
Søren Asmussen ◽  
Jevgenijs Ivanovs
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Phase Type

scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (3) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

scholarly journals Applications of factorization embeddings for Lévy processes

2006 ◽  
Vol 38 (03) ◽  
pp. 768-791 ◽  
Author(s):  
A. B. Dieker
Keyword(s):  
Lévy Process ◽  
Joint Distribution ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Tail Asymptotics ◽  
Lévy Measure ◽  
Risk Models ◽  
General Properties ◽  
Phase Type

We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is ‘attained’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.

An invariant of representations of phase-type distributions and some applications

1996 ◽  
Vol 33 (2) ◽  
pp. 368-381 ◽  
Author(s):  
C. Commault ◽  
J. P. Chemla
Keyword(s):  
Markov Chains ◽  
Laplace Transform ◽  
Rational Functions ◽  
Absorbing State ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Finite State ◽  
The Laplace Transform ◽  
The Difference ◽  
Real Poles

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.

Sign in / Sign up

Export Citation Format

Share Document