Error Bounds for Small Jumps of Lévy Processes

The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

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Applications of factorization embeddings for Lévy processes

Advances in Applied Probability ◽

10.1017/s0001867800001269 ◽

2006 ◽

Vol 38 (03) ◽

pp. 768-791 ◽

Cited By ~ 1

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.

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ORTHOGONAL DECOMPOSITIONS FOR LÉVY PROCESSES WITH AN APPLICATION TO THE GAMMA, PASCAL, AND MEIXNER PROCESSES

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025703001031 ◽

2003 ◽

Vol 06 (01) ◽

pp. 73-102 ◽

Cited By ~ 21

Author(s):

EUGENE LYTVYNOV

Keyword(s):

Brownian Motion ◽

Poisson Process ◽

Lévy Process ◽

Lévy Processes ◽

Random Variable ◽

Levy Processes ◽

Levy Process ◽

Nuclear Space ◽

Multiple Stochastic Integrals ◽

Orthogonal Decompositions

It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. At least three approaches are possible here. The first one, due to Itô, uses the CRP of the Brownian motion and the Poisson process, as well as the representation of a Lévy process through those processes. The second approach, due to Nualart and Schoutens, consists of representing any square-integrable random variable as a sum of multiple stochastic integrals constructed with respect to a family of orthogonalized centered power jumps processes. The third approach, never applied before to the Lévy processes, uses the idea of orthogonalization of polynomials with respect to a probability measure defined on the dual of a nuclear space. The main aims of this paper are to develop the three approaches in the case of a general (ℝ-valued) Lévy process on a Riemannian manifold and (what is more important) to understand a relationship between these approaches. We apply the obtained results to the gamma, Pascal, and Meixner processes, in which case the analysis related to the orthogonalized polynomials becomes essentially simpler and richer than in the general case.

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Construction of a free Lévy process as high-dimensional limit of a Brownian motion on the unitary group

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025715500186 ◽

2015 ◽

Vol 18 (03) ◽

pp. 1550018 ◽

Cited By ~ 1

Author(s):

Michaël Ulrich

Keyword(s):

Brownian Motion ◽

Unitary Group ◽

Lévy Process ◽

Lévy Processes ◽

Levy Processes ◽

Levy Process ◽

Alternative Proof ◽

High Dimensional ◽

General Context ◽

Work Done

It is well known that freeness appears in the high-dimensional limit of independence for matrices. Thus, for instance, the additive free Brownian motion can be seen as the limit of the Brownian motion on hermitian matrices. More generally, it is quite natural to try to build free Lévy processes as high-dimensional limits of classical matricial Lévy processes. We will focus here on one specific such construction, discussing and generalizing the work done previously by Biane in Ref.2, who has shown that the (classical) Brownian motion on the Unitary group U(d) converges to the free multiplicative Brownian motion when d goes to infinity. We shall first recall that result and give an alternative proof for it. We shall then see how this proof can be adapted in a more general context in order to get a free Lévy process on the dual group (in the sense of Voiculescu) U〈n〉. This result will actually amount to a truly noncommutative limit theorem for classical random variables, of which Biane's result constitutes the case n = 1.

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Applications of factorization embeddings for Lévy processes

Advances in Applied Probability ◽

10.1239/aap/1158685001 ◽

2006 ◽

Vol 38 (3) ◽

pp. 768-791 ◽

Cited By ~ 10

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.

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A UNIFYING APPROACH TO FRACTIONAL LÉVY PROCESSES

Stochastics and Dynamics ◽

10.1142/s0219493712500177 ◽

2013 ◽

Vol 13 (02) ◽

pp. 1250017 ◽

Cited By ~ 14

Author(s):

SEBASTIAN ENGELKE ◽

JEANNETTE H. C. WOERNER

Keyword(s):

Brownian Motion ◽

Lévy Process ◽

Lévy Processes ◽

Moving Average ◽

Levy Processes ◽

Levy Process ◽

Integration Theory ◽

Sample Paths ◽

Second Moments ◽

Moving Average Representation

Starting from the moving average representation of fractional Brownian motion, there are two different approaches to constructing fractional Lévy processes in the literature. Applying L2-integration theory, one can keep the same moving average kernel and replace the driving Brownian motion by a pure jump Lévy process with finite second moments. Alternatively, in the framework of alpha-stable random measures, the Brownian motion is replaced by an alpha-stable Lévy process and the exponent in the kernel is reparametrized by H - 1/α. We now provide a unified approach taking kernels of the form [Formula: see text], where γ can be chosen according to the existing moments and the Blumenthal–Getoor index of the underlying Lévy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g., regularity of the sample paths and the semimartingale property.

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PRICING DERIVATIVES ON TWO-DIMENSIONAL LÉVY PROCESSES

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024906003536 ◽

2006 ◽

Vol 09 (02) ◽

pp. 185-197 ◽

Cited By ~ 10

Author(s):

JOSÉ FAJARDO ◽

ERNESTO MORDECKI

Keyword(s):

Brownian Motion ◽

Lévy Process ◽

Lévy Processes ◽

Main Idea ◽

Levy Processes ◽

Levy Process ◽

Two Dimensional ◽

Girsanov’S Theorem ◽

Duality Approach

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to a problem with one Lévy driven stock in an auxiliary market, baptized as "dual market". In this way, we extend the results obtained by Gerber and Shiu [5] for two-dimensional Brownian motion.

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Oscillation of modes of some semi-stable Lévy processes

Nagoya Mathematical Journal ◽

10.1017/s0027763000004682 ◽

1993 ◽

Vol 132 ◽

pp. 141-153 ◽

Cited By ~ 4

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.

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Approximate Wiener--Hopf Factorization and Monte Carlo Methods for Lévy Processes

Theory of Probability and Its Applications ◽

10.1137/s0040585x97t989441 ◽

2019 ◽

Vol 64 (2) ◽

pp. 186-208 ◽

Cited By ~ 2

Author(s):

O. E. Kudryavtsev

Keyword(s):

Monte Carlo ◽

Monte Carlo Methods ◽

Lévy Processes ◽

Levy Processes

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Variance Optimal Stopping for Geometric Lévy Processes

Advances in Applied Probability ◽

10.1017/s0001867800007734 ◽

2015 ◽

Vol 47 (01) ◽

pp. 128-145 ◽

Cited By ~ 3

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.

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