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The estimation of absorption time distributions of Markov jump processes is an important task in various branches of statistics and applied probability. While the time‐homogeneous case is classic, the time‐inhomogeneous case has recently received increased attention due to its added flexibility and advances in computational power. However, commuting sub‐intensity matrices are assumed, which in various cases limits the parsimonious properties of the resulting representation. This paper develops the theory required to solve the general case through maximum likelihood estimation, and in particular, using the expectation‐maximization algorithm. A reduction to a piecewise constant intensity matrix function is proposed in order to provide succinct representations, where a parametric linear model binds the intensities together. Practical aspects are discussed and illustrated through the estimation of notoriously demanding theoretical distributions and real data, from the perspective of matrix analytic methods. [ABSTRACT FROM AUTHOR]

Generalized linear models, introduced by Nelder and Wedderburn, allowed to model the regression of normal and nonnormal data. While doing so, the analysis of these models could not be obtained without the explicit form of the variance function. In this paper, we determine the link and variance functions of the natural exponential family generated by the class of subordinated Lévy processes. In this framework, we introduce a class of variance functions that depends on the Lambert function. In this regard, we call it the Lambert class, which covers the variance functions of the natural exponential families generated by the subordinated gamma processes and the subordinated Lévy processes by the Poisson subordinator. Notice that the gamma process subordinated by the Poisson one is excluded from this class. The concept of reciprocity in natural exponential families was given in order to obtain an exponential family from another one. In this context, we get the reciprocal class of the natural exponential family generated by the class of subordinated Lévy processes. It is well known that the variance function represents an essential element for the determination of the quasi‐likelihood and deviance functions. Then, we use the expression of our variance function in order to maintain them. This leads us to analyze the proposed generalized linear model. We illustrate some of our models with applications to the daily exchange rate returns of the Tunisian Dinar against the U.S. Dollar and the damage incidents of ships. [ABSTRACT FROM AUTHOR]

In this article we present a parametric estimation method for certain multiparameter heavy‐tailed Lévy‐driven moving averages. The theory relies on recent multivariate central limit theorems obtained via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to previous papers, 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 Lévy motion. We extend their work to allow for a multiparametric 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. [ABSTRACT FROM AUTHOR]

In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure‐jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi‐likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we study the detailed distributional properties of integrated non-Gaussian Ornstein–Uhlenbeck (intOU) processes. Both exact and approximate results are given. We emphasize the study of the tail behaviour of the intOU process. Our results have many potential applications in financial economics, as OU processes are used as models of instantaneous variance in stochastic volatility (SV) models. In this case, an intOU process can be regarded as a model of integrated variance. Hence, the tail behaviour of the intOU process will determine the tail behaviour of returns generated by SV models. [ABSTRACT FROM AUTHOR]

For the multivariate COGARCH process, we obtain explicit expressions for the second‐order structure of the "squared returns" process observed on an equidistant grid. Based on this, we present a generalized method of moments estimator for its parameters. Under appropriate moment and strong mixing conditions, we show that the resulting estimator is consistent and asymptotically normal. Sufficient conditions for strong mixing, stationarity and identifiability of the model parameters are discussed in detail. We investigate the finite sample behavior of the estimator in a simulation study. [ABSTRACT FROM AUTHOR]

Let {Zt}t≥0 be a Lévy process with Lévy measure ν and let τ(t) = ∫01 g(...(u) du be a random clock, where g is a non-negative function and {...(t)}≥0 is an ergodic diffusion independent of Z. Time-changed Lévy models of the form Xt:= Zτt are known to incorporate several important stylized features of asset prices, such as leptokurtic distributions and volatility clustering. In this article, we prove central limit theorems for a type of estimators of the integral parameter β( ϕ):=∫ ϕ( x) ν(d x), valid when both the sampling frequency and the observation time-horizon of the process get larger. Our results combine the long-run ergodic properties of the diffusion process with the short-term ergodic properties of the Lévy process Z via central limit theorems for martingale differences. The performance of the estimators are illustrated numerically for Normal Inverse Gaussian process Z and a Cox-Ingersoll-Ross process .... [ABSTRACT FROM AUTHOR]

Lévy processes and infinitely divisible distributions are increasingly defined in terms of their Lévy measure. In order to describe the dependence structure of a multivariate Lévy measure, Tankov (2003) introduced Lévy copulas on . (For an extension to m, see Kallsen & Tankov, 2006 .) Together with the marginal Lévy measures they completely describe multivariate Lévy measures on . In this article we show that any such Lévy copula defines itself a Lévy measure with one-stable margins, in a canonical way. A limit theorem is obtained, characterizing convergence of Lévy measures with the aid of Lévy copulas. Homogeneous Lévy copulas are considered in detail. They correspond to Lévy processes which have a time-constant Lévy copula, and a complete description of homogeneous Lévy copulas is obtained. A general scheme to construct multivariate distributions having special properties is outlined, for distributions with prescribed margins having the same properties. This makes use of Lévy copulas and of certain mappings of Upsilon type. The construction is then exemplified for distributions in the Goldie–Steutel–Bondesson class, the Thorin class and for self-decomposable distributions. [ABSTRACT FROM AUTHOR]