For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?

2021 ◽  
Vol 105 (0) ◽  
pp. 79-91
Author(s):  
F. Kühn ◽  
R. Schilling
Keyword(s):  
Lebesgue Measure ◽  
Lévy Process ◽  
Levy Process ◽  
Content Type ◽  
One Dimensional ◽  
Exponential Moments ◽  
Bounded Functions

Let X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} be a one-dimensional Lévy process such that each X t X_t has a C b 1 C^1_b -density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions f : R → R f\colon \mathbb {R}\to \mathbb {R} , and exponentially bounded functions g : R → ( 0 , ∞ ) g\colon \mathbb {R}\to (0,\infty ) , such that f ( X t ) − E f ( X t ) f(X_t)-\mathbb {E} f(X_t) , resp. g ( X t ) / E g ( X t ) g(X_t)/\mathbb {E} g(X_t) , are martingales.

scholarly journals One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

2009 ◽  
Vol 41 (2) ◽  
pp. 367-392 ◽  
Author(s):  
Shai Covo
Keyword(s):  
Distribution Function ◽  
Lévy Process ◽  
Marginal Distribution ◽  
Levy Process ◽  
Gamma Process ◽  
Lévy Measure ◽  
Original Process ◽  
One Dimensional ◽  
Analogous Formula

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

scholarly journals Pareto Lévy Measures and Multivariate Regular Variation

2012 ◽  
Vol 44 (1) ◽  
pp. 117-138 ◽  
Author(s):  
Irmingard Eder ◽  
Claudia Klüppelberg
Keyword(s):  
Regular Variation ◽  
Lévy Process ◽  
Tail Dependence ◽  
Levy Process ◽  
Dependence Structure ◽  
Lévy Measure ◽  
Multivariate Distributions ◽  
One Dimensional ◽  
The One ◽  
Lévy Measures

We consider regular variation of a Lévy process X := (Xt)t≥0 in with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

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 One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

2009 ◽  
Vol 41 (02) ◽  
pp. 367-392 ◽  
Author(s):  
Shai Covo
Keyword(s):  
Distribution Function ◽  
Lévy Process ◽  
Marginal Distribution ◽  
Levy Process ◽  
Gamma Process ◽  
Lévy Measure ◽  
Original Process ◽  
One Dimensional ◽  
Analogous Formula

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution functionFs(x;t) at timetof the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution functionF(⋅;t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for thenth derivative, ∂nFs(x;t) ∂xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

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.

Pareto Lévy Measures and Multivariate Regular Variation

2012 ◽  
Vol 44 (01) ◽  
pp. 117-138 ◽  
Author(s):  
Irmingard Eder ◽  
Claudia Klüppelberg
Keyword(s):  
Regular Variation ◽  
Lévy Process ◽  
Tail Dependence ◽  
Levy Process ◽  
Dependence Structure ◽  
Lévy Measure ◽  
Multivariate Distributions ◽  
One Dimensional ◽  
The One ◽  
Lévy Measures

We consider regular variation of a Lévy process X := ( X t) t≥0 in with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

scholarly journals On the parabolic generator of a general one-dimensional Lévy process

2008 ◽  
Vol 13 (0) ◽  
pp. 198-209 ◽  
Author(s):  
Nathalie Eisenbaum ◽  
Andreas Kyprianou
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
One Dimensional

scholarly journals Extinction rate of continuous state branching processes in critical Lévy environments

2021 ◽  
Author(s):  
Juan Carlos Pardo ◽  
Vincent Bansaye ◽  
Charline Smadi
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Branching Process ◽  
Branching Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Extinction Rate ◽  
Exponential Moments ◽  
Continuous State ◽  
Remarkable Result

We study  the  speed  of extinction of continuous state branching processes in a Lévy environment, where the associated Lévy process oscillates.  Assuming that the  Lévy process satisfies  Spitzer's condition and the existence of some  exponential moments, we extend recent results where the associated branching mechanism is stable. The study  relies on the  path analysis of  the branching process  together with its Lévy environment, when the latter is conditioned to have a non negative running infimum. For that purpose,  we combine the  approach  developed in    Afanasyev et al. \cite{Afanasyev2005},  for the discrete setting and i.i.d. environments, with fluctuation theory of Lévy processes and a remarkable result on exponential functionals of Lévy processes under Spitzer's condition due to Patie and Savov \cite{patie2016bernstein}.

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