scholarly journals Signal processing with Lévy information

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120433 ◽  
Author(s):  
Dorje C. Brody ◽  
Lane P. Hughston ◽  
Xun Yang
Keyword(s):  
Lévy Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Levy Process ◽  
Inverse Gaussian ◽  
Information Process ◽  
True Value ◽  
Information Processes ◽  
Noise Type ◽  
Exponential Moments

Lévy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Lévy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a ‘message’, then under the transformed measure the original Lévy process takes on the character of an ‘information process’. In this paper we develop a theory of such Lévy information processes. The underlying Lévy process, which we call the fiducial process, represents the ‘noise type’. Each such noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverse Gaussian and normal inverse Gaussian type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations.

Markov Chain Monte Carlo simulation of the distribution of some perpetuities

1999 ◽  
Vol 31 (01) ◽  
pp. 112-134 ◽  
Author(s):  
Jostein Paulsen ◽  
Arne Hove
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Lévy Process ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Levy Process ◽  
Present Value ◽  
Strong Markov

We study the present value Z ∞ = ∫0 ∞ e-X t- dY t where (X,Y) is an integrable Lévy process. This random variable appears in various applications, and several examples are known where the distribution of Z ∞ is calculated explicitly. Here sufficient conditions for Z ∞ to exist are given, and the possibility of finding the distribution of Z ∞ by Markov chain Monte Carlo simulation is investigated in detail. Then the same ideas are applied to the present value Z - ∞ = ∫0 ∞ exp{-∫0 t R s ds}dY t where Y is an integrable Lévy process and R is an ergodic strong Markov process. Numerical examples are given in both cases to show the efficiency of the Monte Carlo methods.

scholarly journals DETERMINATION OF THE LÉVY EXPONENT IN ASSET PRICING MODELS

2019 ◽  
Vol 22 (01) ◽  
pp. 1950008
Author(s):  
GEORGE BOUZIANIS ◽  
LANE P. HUGHSTON
Keyword(s):  
Lévy Process ◽  
Complete Information ◽  
Levy Process ◽  
Pricing Kernel ◽  
Market Prices ◽  
Risky Assets ◽  
Asset Pricing Models ◽  
Benchmark Portfolio ◽  
Exponential Moments

We consider the problem of determining the Lévy exponent in a Lévy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure [Formula: see text], consists of a pricing kernel [Formula: see text] together with one or more non-dividend-paying risky assets driven by the same Lévy process. If [Formula: see text] denotes the price process of such an asset, then [Formula: see text] is a [Formula: see text]-martingale. The Lévy process [Formula: see text] is assumed to have exponential moments, implying the existence of a Lévy exponent [Formula: see text] for [Formula: see text] in an interval [Formula: see text] containing the origin as a proper subset. We show that if the prices of power-payoff derivatives, for which the payoff is [Formula: see text] for some time [Formula: see text], are given at time [Formula: see text] for a range of values of [Formula: see text], where [Formula: see text] is the so-called benchmark portfolio defined by [Formula: see text], then the Lévy exponent is determined up to an irrelevant linear term. In such a setting, derivative prices embody complete information about price jumps: in particular, the spectrum of the price jumps can be worked out from current market prices of derivatives. More generally, if [Formula: see text] for a general non-dividend-paying risky asset driven by a Lévy process, and if we know that the pricing kernel is driven by the same Lévy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the Lévy exponent up to a transformation [Formula: see text], where [Formula: see text] and [Formula: see text] are constants.

scholarly journals A bivariate Lévy process with negative binomial and gamma marginals

2008 ◽  
Vol 99 (7) ◽  
pp. 1418-1437 ◽  
Author(s):  
Tomasz J. Kozubowski ◽  
Anna K. Panorska ◽  
Krzysztof Podgórski
Keyword(s):  
Lévy Process ◽  
Negative Binomial ◽  
Levy Process

scholarly journals Mixed Compound Poisson Distributions

1986 ◽  
Vol 16 (S1) ◽  
pp. S59-S79 ◽  
Author(s):  
Gord Willmot
Keyword(s):  
Negative Binomial ◽  
Random Variable ◽  
Computational Formula ◽  
Inverse Gaussian ◽  
Poisson Distributions ◽  
Mixing Distributions ◽  
Chi Squared ◽  
General Distributions ◽  
Generalized Inverse Gaussian ◽  
Thick Tails

AbstractThe distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous).Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions.It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.

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.

scholarly journals Risk Theory with the Generalized Inverse Gaussian Lévy Process

2004 ◽  
Vol 34 (2) ◽  
pp. 361-377 ◽  
Author(s):  
Manuel Morales
Keyword(s):  
Lévy Process ◽  
Risk Model ◽  
Generalized Inverse ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Inverse Gaussian ◽  
Lévy Measure ◽  
Compound Poisson ◽  
Inverse Gaussian Process ◽  
Generalized Inverse Gaussian

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

scholarly journals On extended stochastic integrals with respect to Lévy processes

2013 ◽  
Vol 5 (2) ◽  
pp. 256-278 ◽  
Author(s):  
N.A. Kachanovsky
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Stochastic Integral ◽  
Random Measure ◽  
Random Variable ◽  
Levy Processes ◽  
Levy Process ◽  
Jump Processes ◽  
Stochastic Integrals ◽  
Stochastic Derivative

Let $L$ be a Levy process on $[0,+\infty)$. In particular cases, when $L$ is a Wiener or Poisson process, any square integrable random variable can be decomposed in a series of repeated stochastic integrals from nonrandom functions with respect to $L$. This property of $L$, known as the chaotic representation property (CRP), plays a very important role in the stochastic analysis. Unfortunately, for a general Levy process the CRP does not hold. There are different generalizations of the CRP for Levy processes. In particular, under the Ito's approach one decomposes a Levy process $L$ in the sum of a Gaussian process and a stochastic integral with respect to a Poisson random measure, and then uses the CRP for both terms in order to obtain a generalized CRP for $L$. The Nualart-Schoutens's approach consists in decomposition of a square integrable random variable in a series of repeated stochastic integrals from nonrandom functions with respect to so-called orthogonalized centered power jump processes, these processes are constructed with using of a cadlag version of $L$. The Lytvynov's approach is based on orthogonalization of continuous monomials in the space of square integrable random variables. In this paper we construct the extended stochastic integral with respect to a Levy process and the Hida stochastic derivative in terms of the Lytvynov's generalization of the CRP; establish some properties of these operators; and, what is most important, show that the extended stochastic integrals, constructed with use of the above-mentioned generalizations of the CRP, coincide.  

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