scholarly journals Wiener-Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms

2008 ◽  
Vol 45 (01) ◽  
pp. 118-134 ◽  
Author(s):  
Alan L. Lewis ◽  
Ernesto Mordecki
Keyword(s):  
Fourier Transform ◽  
Closed Form ◽  
Rational Function ◽  
Ruin Probability ◽  
Lévy Process ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Levy Process ◽  
Type Formula ◽  
Positive Factor

We show that the positive Wiener-Hopf factor of a Lévy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Lévy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.

scholarly journals Wiener-Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms

2008 ◽  
Vol 45 (1) ◽  
pp. 118-134 ◽  
Author(s):  
Alan L. Lewis ◽  
Ernesto Mordecki
Keyword(s):  
Fourier Transform ◽  
Closed Form ◽  
Rational Function ◽  
Ruin Probability ◽  
Lévy Process ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Levy Process ◽  
Type Formula ◽  
Positive Factor

We show that the positive Wiener-Hopf factor of a Lévy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Lévy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.

scholarly journals Conditioned stable Lévy processes and the Lamperti representation

2006 ◽  
Vol 43 (04) ◽  
pp. 967-983 ◽  
Author(s):  
M. E. Caballero ◽  
L. Chaumont
Keyword(s):  
Ruin Probability ◽  
Lévy Process ◽  
Lévy Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Lévy Process ◽  
Self Similar ◽  
Positive Half

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

scholarly journals Conditioned stable Lévy processes and the Lamperti representation

2006 ◽  
Vol 43 (4) ◽  
pp. 967-983 ◽  
Author(s):  
M. E. Caballero ◽  
L. Chaumont
Keyword(s):  
Ruin Probability ◽  
Lévy Process ◽  
Lévy Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Lévy Process ◽  
Self Similar ◽  
Positive Half

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

scholarly journals Oscillation of modes of some semi-stable Lévy processes

1993 ◽  
Vol 132 ◽  
pp. 141-153 ◽  
Author(s):  
Toshiro Watanabe
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Processes ◽  
Unimodal Lévy Process ◽  
Oscillating Mode

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

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.

Variance Optimal Stopping for Geometric Lévy Processes

2015 ◽  
Vol 47 (01) ◽  
pp. 128-145 ◽  
Author(s):  
Kamille Sofie Tågholt Gad ◽  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Maximum Variance ◽  
Geometric Levy Process

The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

Multiple points for symmetric Lévy processes

1978 ◽  
Vol 83 (1) ◽  
pp. 83-90 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Transition Function ◽  
Levy Processes ◽  
Levy Process ◽  
Dimensional Euclidean Space ◽  
Multiple Points ◽  
Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

scholarly journals Necessity of weak subordination for some strongly subordinated Lévy processes

2021 ◽  
Vol 58 (4) ◽  
pp. 868-879
Author(s):  
Boris Buchmann ◽  
Kevin W. Lu
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process

AbstractConsider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.

scholarly journals A Liouville theorem for Lévy generators

Positivity ◽  
2020 ◽  
Author(s):  
Franziska Kühn
Keyword(s):  
Weak Solution ◽  
Laplace Equation ◽  
Lévy Process ◽  
Lévy Processes ◽  
Infinitesimal Generator ◽  
Polynomial Growth ◽  
Liouville Theorem ◽  
Levy Processes ◽  
Levy Process ◽  
Regularity Estimates

AbstractUnder mild assumptions, we establish a Liouville theorem for the “Laplace” equation $$Au=0$$ A u = 0 associated with the infinitesimal generator A of a Lévy process: If u is a weak solution to $$Au=0$$ A u = 0 which is at most of (suitable) polynomial growth, then u is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes.

scholarly journals Notes on energy for space-time processes over Lévy processes

1991 ◽  
Vol 122 ◽  
pp. 63-74 ◽  
Author(s):  
Mamoru Kanda
Keyword(s):  
Euclidean Space ◽  
Probability Measure ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Space Time ◽  
Levy Process ◽  
Independent Increments

Let X = (Xt, 0 ≤ t < ∞) be a Lévy process on the Euclidean space Rd, that is, a process on Rd with stationary independent increments which has right continuous paths with left limits. We denote by Px the probability measure such that PX(X0 = x) = 1 and by Ex the expectation relative to Px. The process is characterized by the exponent Ψ through.

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