On Error Rates in Normal Approximations and Simulation Schemes for Lévy Processes

Let X=(X(t) : t≥0) be a Lévy process. In simulation, one often wants to know at what size it is possible to truncate the small jumps while retaining enough accuracy. A useful tool here is the Edgeworth expansion. We provide a third order expansion together with a uniform error bound, assuming third Lévy moment is 0. We next discuss approximating X in the finite variation case. Truncating the small jumps, we show that, adding their expected value, and further, including their variability by approximating by a Brownian motion, gives successively better results in general. Finally, some numerical illustrations involving a normal inverse Gaussian Lévy process are given.     

Lévy Processes,Phase-Type Distributions,and Martingales

Lévy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance, etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance gamma processes, CGMY Lévy processes (tempered stable processes), NIG (normal inverse Gaussian) Lévy processes, and hyperbolic Lévy processes. We consider here a dense class of Lévy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of Lévy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a Lévy process with reflection in 0. The solutions are in terms of roots to polynomials, and the basic equations are derived by purely probabilistic arguments using martingale optional stopping; a particularly useful martingale is the so-called Kella-Whitt martingale. Also, the relation to fluid models with a Brownian component is discussed.     

The Discounted Berry-Esséen Analogue for Autoregressive Processes

In this article, the discounted Berry-Esséen analogs and non uniform estimates for autoregressive processes are obtained.     

A Semi-parametric Bayesian Analysis of Survival Data Based on Lévy-driven Processes

In the presence of covariate information, the proportional hazards model is one of the most popular models. In this paper, in a Bayesian nonparametric framework, we use a Markov (Lévy-driven) process to model the baseline hazard rate. Previous Bayesian nonparametric models have been based on neutral to the right processes, which have a number of drawbacks, such as discreteness of the cumulative hazard function. We allow the covariates to be time dependent functions and develop a full posterior analysis via substitution sampling. A detailed illustration is presented.     

Rejection sampling for tempered Lévy processes

Statistics and Computing - We extend the idea of tempering stable Lévy processes to tempering more general classes of Lévy processes. We show that the original process can be decomposed...     

On Some Applications of a Symbolic Representation of Non Centered Lévy Processes

By using a symbolic technique known in the literature as the classical umbral calculus, we characterize two classes of polynomials related to Lévy processes: the Kailath-Segall and the time-space harmonic polynomials. We provide the Kailath-Segall formula in terms of cumulants and we recover simple closed-forms for several families of polynomials with respect to not centered Lévy processes, such as the Hermite polynomials with Brownian motion, Poisson-Charlier polynomials with Poisson processes, actuarial polynomials with Gamma processes, first kind Meixner polynomials with Pascal processes, and Bernoulli, Euler, and Krawtchuk polynomials with suitable random walks.     

Shrinkage Estimation of the Memory Parameter in Stationary Gaussian Processes

The correct and efficient estimation of memory parameters in a stationary Gaussian processes is an important issue, since otherwise, forecasts based on the resulting time series would be misleading. On the other hand, if the memory parameters are suspected to fall in a smaller subspace through some hypothesis restrictions, it becomes a hard decision whether to use estimators based on the restricted spaces or to use unrestricted estimators over the full parameter space. In this article, we propose James-Stein-type estimators of the memory parameters of a stationary Gaussian times series process, which can efficiently incorporate the hypothetical restrictions. We show theoretically that the proposed estimators are more efficient than the usual unrestricted maximum likelihood estimators over the entire parameter space.     

Multilevel particle filters for Lévy-driven stochastic differential equations

Statistics and Computing - We develop algorithms for computing expectations with respect to the laws of models associated to stochastic differential equations driven by pure Lévy processes. We...     

The estimation for Lévy processes in high frequency data

In this article, a generalized Lévy model is proposed and its parameters are estimated in high-frequency data settings. An infinitesimal generator of Lévy processes is used to study the asymptotic properties of the drift and volatility estimators. They are consistent asymptotically and are independent of other parameters making them better than those in Chen et al. (2010 Chen, S. X., Delaigle, A., Hall, P. (2010). Nonparametric estimation for a class of Lévy processes. Journal of Econometrics 157:257–271.[Crossref], [Web of Science ®] , [Google Scholar]). The estimators proposed here also have fast convergence rates and are simple to implement.     

Locally most powerful two-sample rank tests for Lévy distributions

For two independent samples of independent random variables which follow a Lévy distribution, the scores for the locally most powerful rank tests for the location and scale problem are obtained. To carry the asymptotic normality of the rank statistics into practice the null means and variances are calculated. Research supported by Deutsche Forschungsgemeinschaft (DFG).     

A Correlation Test for Normality Based on the Lévy Characterization

A powerful test of fit for normal distributions is proposed. Based on the Lévy characterization, the test statistic is the sample correlation coefficient of normal quantiles and sums of pairs of observations from a random sample. Since the test statistic is location-scale invariant, critical values can be obtained by simulation without estimating any parameters. It is proved that this test is consistent. A power comparison study including some directed tests shows that the proposed test is competitive, it is more powerful than the well-known Jarque–Bera test, and it is comparable to Shapiro–Wilk test against a number of alternatives.     

On some dependence structures for multidimensional Lévy driven moving averages

The Lévy copula can describe the dependence structure of a multidimensional Lévy process or a multivariate infinitely divisible random variable. Suppose the Lévy copula of a multidimensional Lévy process is known. We present the Lévy copula of the Lévy measure of the moving average driven by the multidimensional Lévy process. If there exist some special dependence structures among the components of the Lévy process, we give some dependence invariance properties after the transform of the moving average.     

Stepanov-like almost automorphic solutions for stochastic differential equations with Lévy noise

In this paper, we introduce a new concept of Poisson Stepanov-like almost automorphy (or Poisson S²-almost automorphy). Under some suitable conditions on the coefficients, we establish the existence and uniqueness of Stepanov-like almost automorphic mild solution to a class of semilinear stochastic differential equations with infinite dimensional Lévy noise. We further discuss the global asymptotic stability of these solution. Finally, we give an example to illustrate the theoretical results obtained in this paper.     

Correction to: Multilevel particle filters for Lévy-driven stochastic differential equations

Statistics and Computing - The article Multilevel particle filters for Lévy-driven stochastic differential equations, written by Ajay Jasra, Kody J. H. Law, Prince Peprah Osei, was originally...     

An application of Lévy's inversion formula for higher order asymptotic expansion

The asymptotic expansions for the distribution of statistics are, in general, given by applying Lévy's inversion formula to the characteristic function. This paper shows an inversion formula for higher order asymptotic expansion of the distribution of a scalar valued function which contains dependent statistics. The usage of the formula is illustrated by derivation of third order asymptotic expansion of the distribution of Hotelling's T²-statistic under the elliptical distribution as an example.     

Estimation procedures for grouped data – a comparative study

Interval-censored data are very common in the reliability and lifetime data analysis. This paper investigates the performance of different estimation procedures for a special type of interval-censored data, i.e. grouped data, from three widely used lifetime distributions. The approaches considered here include the maximum likelihood estimation, the minimum distance estimation based on chi-square criterion, the moment estimation based on imputation (IM) method and an ad hoc estimation procedure. Although IM-based techniques are extensively used recently, we show that this method is not always effective. It is found that the ad hoc estimation procedure is equivalent to the minimum distance estimation with another distance metric and more effective in the simulation. The procedures of different approaches are presented and their performances are investigated by Monte Carlo simulation for various combinations of sample sizes and parameter settings. The numerical results provide guidelines to analyse grouped data for practitioners when they need to choose a good estimation approach.     

Semi-Parametric Estimation for Forward–Backward Stochastic Differential Equations

In this article, we suggest a semi-parametric estimation for Forward–Backward Stochastic Differential Equations (FBSDE), with a linear generator. Both the nonparametric and parametric estimators are computationally feasible and the asymptotic properties are standard in the sense of normality. Although there is a plug-in nonparametric estimator in parametric estimation, the high order kernel, under-smoothing and bias correction are not required. Some simulation studies are also given to illustrate our methods.     

Tempered Mittag-Leffler Lévy processes

In this article, we introduce tempered Mittag-Leffler Lévy processes (TMLLP). TMLLP is represented as tempered stable subordinator delayed by a gamma process. Its probability density function and Lévy density are obtained in terms of infinite series and Mittag-Leffler function, respectively. Asymptotic forms of the tails and moments are given. A step-by-step procedure of the parameters estimation and simulation of sample paths is given. We also provide main results available for Mittag-Leffler Lévy processes (MLLP) and some extensions which are not available in a collective way in a single article. Our results generalize and complement the results available on Mittag-Leffler distribution and MLLP in several directions. Further, the asymptotic forms of the moments of the first-exit times of the TMLLP are also discussed.     

Option pricing for a stochastic volatility Lévy model with stochastic interest rates

An alternative option pricing model under a forward measure is proposed, in which asset prices follow a stochastic volatility Lévy model with stochastic interest rate. The stochastic interest rate is driven by the Hull–White process. By using an approximate method, we find a formulation for the European option in term of the characteristic function of the tail probabilities.