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A novel theoretical result on estimation of the local time and the occupation time measure of an α-stable Lévy process with α ∈ (1, 2) is presented. The approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an L²-optimal statistic by construction. The corresponding stable central limit theorems are proved and a statistical application is discussed. In particular, this work extends the results of [20], which investigated the case of the Brownian motion. [ABSTRACT FROM AUTHOR]

This paper provides a multivariate extension of Bertoin's pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged. [ABSTRACT FROM AUTHOR]

In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477-4528, Festschrift for Bernt Øksendal, Stochastics 81(1) (2017), 360-383] under the assumption that the kernel function potentially exhibits a singular behaviour at 0. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article [Stoch. Process. Appl. 125(2) (2014), 653-677] that studied the same mathematical question for the class of Brownian semi-stationary models. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the laws of exponential functionals of the processes X = (Xs)s≥0 with independent increments, namely It = ƒt0 exp(-Xs)ds, t ≥ 0, and also I∞ = ƒ∞0 exp(-Xs)ds. Under suitable conditions, the integro-differential equations for the density of It and I8 are derived. Sufficient conditions are derived for the existence of a smooth density of the laws of these functionals with respect to the Lebesgue measure. In the particular case of Lévy processes these equations can be simplified and, in a number of cases, solved explicitly. [ABSTRACT FROM AUTHOR]

The existence of density function of the running maximum of a stochastic differential equation (SDE) driven by a Brownian motion and a nontruncated pure-jump process is verified. This is proved by the existence of density function of the running maximum of the Wiener–Poisson functionals resulting from Bismut’s approach to the Malliavin calculus for jump processes. [ABSTRACT FROM AUTHOR]

Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric Lévy measures and the corresponding real-valued continuous negative definite functions. The Lévy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework. From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance Multivariance: New dependence measures for random vectors (submitted). [ABSTRACT FROM AUTHOR]

In this article, a non-Gaussian long memory process is constructed by the aggregation of independent copies of a fractional Lévy Ornstein-Uhlenbeck process with random coefficients. Several properties and a limit theorem are studied for this new process. Finally, some simulations of the limit process are shown. [ABSTRACT FROM AUTHOR]

In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477–4528, Festschrift for Bernt Øksendal, Stochastics 81(1) (2017), 360–383] under the assumption that the kernel function potentially exhibits a singular behaviour at 0. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article [Stoch. Process. Appl. 125(2) (2014), 653–677] that studied the same mathematical question for the class of Brownian semi-stationary models.

A new modified Φ-Sobolev inequality for canonical L²-Lévy processes, which are hybrid cases of the Brownian motion and pure jump-Lévy processes, is developed. Existing results included only a part of the Brownian motion process and pure jump processes. A generalized version of the Φ-Sobolev inequality for the Poisson and Wiener spaces is derived. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical Lévy processes. In contrast to the measure concentration inequalities for the Brownian motion alone or pure jump Lévy processes alone, the measure concentration inequalities for canonical Lévy processes involve Lambert's W-function. Examples of inequalities are also presented, such as the supremum of Lévy processes in the case of mixed Brownian motion and Poisson processes. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for Lévy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary Lévy process.

Let {L(t), t ≥ 0} be a Lévy process with representative random variable L(1) defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy measure of this process. We also define two subordinated processes. The first one, Y(t), is a Negative-Binomial process X(t) directed by Gamma process. The second process, Z(t), is a Logarithmic Lévy process L(t) directed by Poisson process. For them, we prove that the Bernstein functions of the processes L(t) and Y(t) contain the iterated logarithmic function. In addition, the Lévy measure of the subordinated process Z(t) is a shifted Lévy measure of the Negative-Binomial process X(t).We compare the properties of these processes, knowing that the total masses of corresponding Lévy measures are equal. [ABSTRACT FROM AUTHOR]

We investigate the pricing of cliquet options in a jump-diffusion model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a drifted Lévy process entailing a Brownian diffusion component as well as compound Poisson jumps. We also derive representations for the density and distribution function of the emerging Lévy process. In this setting, we infer semi-analytic expressions for the cliquet option price by two different approaches. The first one involves the probability distribution function of the driving Lévy process whereas the second draws upon Fourier transform techniques. With view on sensitivity analysis and hedging purposes, we eventually deduce representations for several Greeks while putting emphasis on the Vega. [ABSTRACT FROM AUTHOR]

We investigate the pricing of cliquet options in a geometric Meixner model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a pure-jump Meixner-Lévy process yielding Meixner distributed log-returns. In this setting, we infer semi-analytic expressions for the cliquet option price by using the probability distribution function of the driving Meixner-Lévy process and by an application of Fourier transform techniques. In an introductory section, we compile various facts on the Meixner distribution and the related class of Meixner-Lévy processes. We also propose a customized measure change preserving the Meixner distribution of any Meixner process. [ABSTRACT FROM AUTHOR]