SPLITTING OF POISSON NOISE AND LÉVY PROCESSES ON REAL LIE ALGEBRAS

2002 ◽  
Vol 05 (01) ◽  
pp. 21-40 ◽  
Author(s):  
NICOLAS PRIVAULT
Keyword(s):  
Finite Difference ◽  
Poisson Process ◽  
Lie Algebras ◽  
Discrete Time ◽  
Lévy Processes ◽  
Levy Processes ◽  
Poisson Noise ◽  
Probabilistic Interpretation ◽  
Time Changes ◽  
Binomial Process

The compensated Poisson noise is expressed as a composite sum (splitting) of creation and annihilation operators, whose probabilistic interpretation relies on time changes. We construct an Itô table for this decomposition and obtain continuous and discrete time realizations of Lévy processes on the finite difference algebra [Formula: see text] and on [Formula: see text], e.g. the space–time dual of the Poisson process (compensated gamma process), and the continuous binomial process.

QUASI-INVARIANCE FORMULAS FOR COMPONENTS OF QUANTUM LÉVY PROCESSES

2004 ◽  
Vol 07 (01) ◽  
pp. 131-145 ◽  
Author(s):  
UWE FRANZ ◽  
NICOLAS PRIVAULT
Keyword(s):  
Lie Algebras ◽  
Lévy Processes ◽  
Unitary Operator ◽  
Levy Processes ◽  
Absolutely Continuous ◽  
Commutative Subalgebra ◽  
Independent Increments ◽  
Commutative Subalgebras ◽  
Current Representation ◽  
General Method

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,15,16 and the Meixner process.

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

2003 ◽  
Vol 06 (01) ◽  
pp. 73-102 ◽  
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.

Time Changes for Lévy Processes

2001 ◽  
Vol 11 (1) ◽  
pp. 79-96 ◽  
Author(s):  
Hélyette Geman ◽  
Dilip B. Madan ◽  
Marc Yor
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Time Changes

scholarly journals Finite Difference Methods for Option Pricing under Lévy Processes: Wiener-Hopf Factorization Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Oleg Kudryavtsev
Keyword(s):  
Finite Difference ◽  
Lévy Processes ◽  
Finite Difference Methods ◽  
Factorization Method ◽  
Levy Processes ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Pricing Options ◽  
Black Scholes ◽  
Difference Methods

In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments.

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