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In this paper we present a parametric estimation method for certain multi-parameter heavy-tailed Levy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained in [3] via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to the papers [14, 15], which propose to use the marginal empirical characteristic function to estimate the one-dimensional parameter of the kernel function and the stability index of the driving Levy motion. We extend their work to allow for a multi-parametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck process, certain CARMA(2, 1) models and Ornstein-Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

Let $(Q_t)$ be a stationary workload process, and $r(t)$ the correlation coefficient of $Q_0$ and $Q_t$. In a series of previous papers (i) the transform of $r(\cdot)$ has been derived for the case that the driving process is spectrally-positive (sp) or spectrally-negative (sn) Levy, (ii) it has been shown that for sp-Levy and sn-Levy input $r(\cdot)$ is positive, decreasing, and convex, (iii) in case the driving Levy process is light-tailed (a condition that is automatically fulfilled in the sn case), the decay of the decay rate agrees with that of the tail of the busy period distribution. In the present paper we first prove the conjecture that property (ii) carries over to spectrally two-sided Levy processes; we do so for the case the Levy process is reflected at 0, and the case it is reflected at 0 and $K > 0$. Then we focus on queues fed by Markov additive processes (maps). We start by the establishing the counterpart of (i) for sp- and sn-maps. Then we refute property (ii) for maps: we construct examples in which the correlation coefficient can be (locally) negative, decreasing, and concave. Finally, in relation to (iii), we point out how to identify the decay rate of $r(\cdot)$ in the light-tailed map case, thus showing that the tail behavior of $r(\cdot)$ does not necessarily match that of the busy-period tail; singularities related to the transition rate matrix of the background Markov chain turn out to play a crucial role here., 20 pages

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper, we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving average and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit. The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving average and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.

In this paper we present some new limit theorems for power variations of stationary increment L\'{e}vy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477--4528, Festschrift for Bernt {\O}ksendal, 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., Comment: Published at https://doi.org/10.15559/18-VMSTA111 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.

In the present paper, we study the chaotic representation property for certain families ${\mathscr{X}}$ of square integrable martingales on a finite time interval $[0,T]$. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family ${\mathscr{X}}$ of square integrable martingales having deterministic mutual predictable covariation $\langle X,Y\rangle$ for all $X,Y\in{\mathscr{X}}$. The main result of the present paper is stated in Theorem 5.8 below: If ${\mathscr{X}}$ is a compensated-covariation stable family of square integrable martingales such that $\langle X,Y\rangle$ is deterministic for all $X,Y\in{\mathscr{X}}$ and, furthermore, the system of monomials generated by ${\mathscr{X}}$ is total in $L^{2}(\Omega,\mathscr{F}^{\mathscr{X}}_{T},\mathbb{P})$, then ${\mathscr{X}}$ possesses the chaotic representation property with respect to the $\sigma$-field $\mathscr{F}^{\mathscr{X}}_{T}$. We shall apply this result to the case of Lévy processes. Relative to the filtration $\mathbb{F}^{L}$ generated by a Lévy process $L$, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Lévy processes, several examples of concrete families ${\mathscr{X}}$ of martingales including Teugels martingales.

We consider the problem of pricing American options in the framework of a well-known stochastic volatility model with jumps, the Bates model. According to this model the asset price is described by a jump-diffusion stochastic differential equation in which the jump term consists of a Lvy process of compound Poisson type, while the volatility is modeled as a CIR-type process correlated with the asset price. Pricing American options under the Bates model requires us to solve a partial integro-differential equation with the final condition and boundary conditions prescribed on a free boundary. In this paper a numerical method for solving such a problem is proposed. In particular, first of all, using a Richardson extrapolation technique, the problem is reduced to a problem with fixed boundary. Then the problem obtained is solved using an ad hoc finite element method which efficiently combines an implicit/explicit time stepping, an operator splitting technique, and a non-uniform mesh of right-angled triangles. Numerical experiments are presented showing that the option pricing algorithm developed in this paper is extremely accurate and fast. In particular it is significantly more efficient than other numerical methods that have recently been proposed for pricing American options under the Bates model. © 2010 Elsevier Ltd. All rights reserved.

In this paper we introduce and study a regularizing one-to-one mapping ϒ 0 from the class of one-dimensional Levy measures into itself. This mapping appeared implicitly in our previous paper [O.E. Barndorff-Nielsen, S. Thorbjornsen, A connection between free and classical infinite divisibility, Inf. Dim. Anal. Quant. Probab. 7 (2004) 573–590], where we introduced a one-to-one mapping ϒ from the class ID ( * ) of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of ϒ 0 in the present paper, we deduce further properties of ϒ . In particular it is proved that ϒ maps the class L ( * ) of selfdecomposable laws onto the so called Thorin class T ( * ) . Further, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family ( ϒ α ) α ∈ [ 0 , 1 ] of one-to-one mappings ϒ α : ID ( * ) → ID ( * ) , which interpolates smoothly between ϒ ( α = 0 ) and the identity mapping on ID ( * ) ( α = 1 ). We prove that each of the mappings ϒ α shares many of the properties of ϒ . In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.

Let X be a spectrally negative self-similar Markov process with 0 as an absorbing state. In this paper, we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density. We provide a power series and a contour integral representation of this density. Then, by means of probabilistic arguments, we deduce some interesting analytical properties satisfied by these functions, which include, for instance, several types of hypergeometric functions. We also give several characterizations of the Kesten's constant appearing in the study of the asymptotic tail distribution of the absorbtion time. We end the paper by detailing some known and new examples. In particular, we offer an alternative proof of the recent result obtained by Bernyk, Dalang and Peskir [Ann. Probab. 36 (2008) 1777--1789] regarding the law of the maximum of spectrally positive L\'{e}vy stable processes., Comment: Published in at http://dx.doi.org/10.1214/10-AOP638 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In the present paper we present a finite element approach to option pricing in the framework of a well-known stochastic volatility model with jumps, the Bates model. In this model the asset log-returns are assumed to follow a jump-diffusion model where the jump component consists of a Levy process of compound Poisson type, while the volatility dynamics is described by a stochastic differential equation of CIR type, with a mean-reverting drift term and a diffusion component correlated with that of the log-returns. Like in all the Levy models, the option pricing problem can be formulated in terms of an integro-differential equation: for the Bates model the unknown C(S, V, t) (the option price) of the pricing equation depends on three independent variables and the differential operator part turns out to be of parabolic kind, while the nonlocal integral operator is calculated with respect to the Levy measure of the jumps. In this paper we will present a variational formulation of the problem suitable for a finite element approach. After evaluating European options as a benchmark, the method will be applied to Barrier (Down and Out) options.

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.

It is common practice to treat small jumps of L\'evy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of L\'evy processes with infinite L\'evy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a L\'evy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of L\'evy processes are provided. The theory is illustrated by concrete examples., Comment: Important and necessary changes have been made in this new version, this version supersedes version 1

In this paper, we study the cut-off phenomenon under the total variation distance of $d$-dimensional Ornstein-Uhlenbeck processes which are driven by L\'evy processes. That is to say, under the total variation distance, there is an abrupt convergence of the aforementioned process to its equilibrium, i.e. limiting distribution. Despite that the limiting distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases under suitable conditions on the limiting distribution. The cut-off phenomena for the average and superposition processes are also determined., Comment: 34 pages. Electronic Journal of Probability 2020

This paper is devoted to the analysis of an optimal control problem for stochastic integro-differential equations driven by a non-gaussian Levy noise. The memory effect in the equation is driven by a completely monotone kernel (thus covering, for instance, the class of fractional time derivative of order less than 1). We suppose that the control acts on the jump rate of the noise. We show that this allows to tackle the problem through a backward stochastic differential equations approach, since the structure condition required by this approach is naturally satisfied. We solve the optimal control problem of minimizing a cost functional on a finite time horizon, with both running and final costs. We finally prove the existence of a weak solution of the closed-loop equation and we construct an optimal feedback control.

This paper is concerned with asymptotic behavior (at zero and at infinity) of the favorite points of Levy processes. By exploring Molchan’s idea for deriving lower tail probabilities of Gaussian processes with stationary increments, we extend the result of Marcus (J Theor Probab 14(3):867–885, 2001) on the favorite points to a larger class of symmetric Levy processes.

In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter $\theta$. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $\Delta$n = sup i=0,...,n--1 (t n i+1 -- t n i) $\rightarrow$ 0. We introduce an estimator of $\theta$, based on a contrast function, which is efficient without requiring any conditions on the rate at which $\Delta$n $\rightarrow$ 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n$\Delta$ 3 n $\rightarrow$ 0 was needed (see [10],[24]) and where the process was supposed to have summable jumps. Moreover, in the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of $\theta$ is feasible under the condition that n$\Delta$ k n $\rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. L{\'e}vy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.

International audience; In the modern world of "Big Data," dynamic signals are often multivariate and characterized by joint scale-free dynamics (self-similarity) and non-Gaussianity. In this paper, we examine the performance of joint wavelet eigenanalysis estimation for the Hurst parameters (scaling exponents) of non-Gaussian multivariate processes. We propose a new process called operator fractional Lévy motion (ofLm) as a Lévy-type model for non-Gaussian multivariate self-similarity. Based on large size Monte Carlo simulations of bivariate ofLm with a combination of Gaussian and non-Gaussian marginals, the estimation performance for Hurst parameters is shown to be satisfactory over finite samples.

In this paper, we study the dynamic risk measures for processes induced by backward stochastic differential equations driven by Teugel’s martingales associated with Lévy processes (BSDELs). The representation theorem for generators of BSDELs is provided. Furthermore, the time consistency of the coherent and convex dynamic risk measures for processes is characterized by means of the generators of BSDELs. Moreover, the coherency and convexity of dynamic risk measures for processes are characterized by the generators of BSDELs. Finally, we provide two numerical examples to illustrate the proposed dynamic risk measures.

This paper is about the structure of all entrance laws (in the sense of Dynkin) for time-inhomogeneous Ornstein–Uhlenbeck processes with Lévy noise in Hilbert state spaces. We identify the extremal entrance laws with finite weak first moments through an explicit formula for their Fourier transforms, generalizing corresponding results by Dynkin for Wiener noise and nuclear state spaces. We then prove that an arbitrary entrance law with finite weak first moments can be uniquely represented as an integral over extremals. It is proved that this can be derived from Dynkin’s seminal work “Sufficient statistics and extreme points” in Ann. Probab. 1978, which contains a purely measure theoretic generalization of the classical analytic Krein–Milman and Choquet Theorems. As an application, we obtain an easy uniqueness proof for [Formula: see text]-periodic entrance laws in the general periodic case. A number of further applications to concrete cases are presented.

In this paper, we study the ruin problem with investment in a general framework where the business part X is a L{\'e}vy process and the return on investment R is a semimartingale. We obtain upper bounds on the finite and infinite time ruin probabilities that decrease as a power function when the initial capital increases. When R is a L{\'e}vy process, we retrieve the well-known results. Then, we show that these bounds are asymptotically optimal in the finite time case, under some simple conditions on the characteristics of X. Finally, we obtain a condition for ruin with probability one when X is a Brownian motion with negative drift and express it explicitly using the characteristics of R.

This work is devoted to the study of semimartingales on the dual of a general nuclear space. We start by establishing conditions for a cylindrical semimartingale in the strong dual Φ′ of a nuclear space Φ to have a Φ′-valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples are also provided. Later, we will show that under some general conditions every semimartingale taking values in the dual of a nuclear space has a canonical representation. The concept of predictable characteristics is introduced and is used to establish necessary and sufficient conditions for a Φ′-valued semimartingale to be a Φ′-valued Lévy process. UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigaciones en Matemáticas Puras y Aplicadas (CIMPA) UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemática

This paper analyses the impact of parametric timing portfolio strategies on the U.S. stock market. In particular, we assume that the log-returns follow a given parametric Levy process and we describe a methodology to approximate the distributions of stopping times using the underlying Markov transition matrix. Therefore, we propose the use of portfolio strategies based on the maximization of the ratio between the expected first passage time to reach a low level of wealth and the expected first passage time to reach a high level of wealth. Finally, we compare the ex-post wealth obtained maximizing the ratio of proper expected stopping times under different distributional assumptions.

In this paper, we establish a large deviation principle for a type of stochastic partial differential equations (SPDEs) with locally monotone coefficients driven by L\'evy noise. The weak convergence method plays an important role., Comment: arXiv admin note: text overlap with arXiv:1203.4020 by other authors