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A Ciesielski–Taylor type identity for positive self-similar Markov processes

A. E. Kyprianou, P. Patie (2011)

Annales de l'I.H.P. Probabilités et statistiques

The aim of this note is to give a straightforward proof of a general version of the Ciesielski–Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski–Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly, a new transformation which maps a subset of the family of Laplace exponents of spectrally negative Lévy processes into itself. Secondly, some classical features...

A fuzzy approach to option pricing in a Levy process setting

Piotr Nowak, Maciej Romaniuk (2013)

International Journal of Applied Mathematics and Computer Science

In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets. We assume that some parameters of the financial instrument cannot be precisely described and therefore they are...

A remarkable σ -finite measure unifying supremum penalisations for a stable Lévy process

Yuko Yano (2013)

Annales de l'I.H.P. Probabilités et statistiques

The σ -finite measure 𝒫 sup which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s h -transform processes with respect to these functions are utilized for the construction of 𝒫 sup .

A simple approach to functional inequalities for non-local Dirichlet forms

Jian Wang (2014)

ESAIM: Probability and Statistics

With direct and simple proofs, we establish Poincaré type inequalities (including Poincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities), entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. The proofs are efficient for non-local Dirichlet forms with general jump kernel, and also work for Lp(p> 1) settings. Our results yield a new sufficient condition for fractional Poincaré inequalities, which were recently studied in [P.T. Gressman,...

Almost automorphic solution for some stochastic evolution equation driven by Lévy noise with coefficients S2−almost automorphic

Mamadou Moustapha Mbaye (2016)

Nonautonomous Dynamical Systems

In this work we first introduce the concept of Poisson Stepanov-like almost automorphic (Poisson S2−almost automorphic) processes in distribution. We establish some interesting results on the functional space of such processes like an composition theorems. Next, under some suitable assumptions, we establish the existence, the uniqueness and the stability of the square-mean almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise in case...

Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders

Z. Michna (2007)

Studia Mathematica

In this paper we consider a symmetric α-stable Lévy process Z. We use a series representation of Z to condition it on the largest jump. Under this condition, Z can be presented as a sum of two independent processes. One of them is a Lévy process Y x parametrized by x > 0 which has finite moments of all orders. We show that Y x converges to Z uniformly on compact sets with probability one as x↓ 0. The first term in the cumulant expansion of Y x corresponds to a Brownian motion which implies that Y x can...

Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2007)

ESAIM: Probability and Statistics

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time Tx of (Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the overshoot and Lx := x- XTx- the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx,Lx) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that ( T x ˜ , K x , L x ) converges in distribution as x → ∞, where T x ˜ denotes a suitable renormalization of Tx.


Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes

Bernard Roynette, Pierre Vallois, Agnès Volpi (2008)

ESAIM: Probability and Statistics

Let ( X t , t 0 ) be a Lévy process started at 0 , with Lévy measure ν . We consider the first passage time T x of ( X t , t 0 ) to level x > 0 , and K x : = X T x - 𝑥 the overshoot and L x : = x - X T 𝑥 - the undershoot. We first prove that the Laplace transform of the random triple ( T x , K x , L x ) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that ( T x ˜ , K x , L x ) converges in distribution as x , where T x ˜ denotes a suitable renormalization of T x .

Asymptotic properties of power variations of Lévy processes

Jean Jacod (2007)

ESAIM: Probability and Statistics

We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Lévy process between the successive times iΔn for i = 0,1,...,n. One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/or centering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen...

Boundary potential theory for stable Lévy processes

Paweł Sztonyk (2003)

Colloquium Mathematicae

We investigate properties of harmonic functions of the symmetric stable Lévy process on d without the assumption that the process is rotation invariant. Our main goal is to prove the boundary Harnack principle for Lipschitz domains. To this end we improve the estimates for the Poisson kernel obtained in a previous work. We also investigate properties of harmonic functions of Feynman-Kac semigroups based on the stable process. In particular, we prove the continuity and the Harnack inequality for...

Characterization of unitary processes with independent and stationary increments

Lingaraj Sahu, Kalyan B. Sinha (2010)

Annales de l'I.H.P. Probabilités et statistiques

This is a continuation of the earlier work (Publ. Res. Inst. Math. Sci.45 (2009) 745–785) to characterize unitary stationary independent increment gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with technical assumptions on the domain of the generator, unitary equivalence of the process to the solution of an appropriate Hudson–Parthasarathy equation is proved.

Convex rearrangements of Lévy processes

Youri Davydov, Emmanuel Thilly (2007)

ESAIM: Probability and Statistics

In this paper we study asymptotic behavior of convex rearrangements of Lévy processes. In particular we obtain Glivenko-Cantelli-type strong limit theorems for the convexifications when the corresponding Lévy measure is regularly varying at + with exponent α ∈ (1,2).

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