Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales

We derive Central Limit Theorems for the convergence of approximate quadratic variations, computed on the basis of regularly spaced observation times of the underlying process, toward the true quadratic variation. This problem was solved in the case of an Itô semimartingale having a non-vanishing continuous martingale part. Here we focus on the case where the continuous martingale part vanishes and find faster rates of convergence, as well as very different limiting processes.     

Strong approximation of semimartingales and statistical processes

Summary As an application of general convergence results for semimartingales, exposed in their book Limit Theorems for Stochastic Processes, Jacod and Shiryaev obtained a fundamental result on the convergence of likelihood ratio processes to a Gaussian limit. We strengthen this result in a quantitative sense and show that versions of the likelihood ratio processes can be defined on the space of the limiting experiment such that we get pathwise almost sure approximations with respect to the uniform metric. The approximations are considered under both sequences of measures, the hypothesisP ⁿ and the alternative . A consequence is e.g. an estimate for the speed of convergence in the Prohorov metric. New approximation techniques for stochastic processes are developed.This article was processed by the author using the L^AT_EX style filepljourIm from Springer-Verlag.     

Limit theorems for bipower variation of semimartingales

This paper presents limit theorems for certain functionals of semimartingales observed at high frequency. In particular, we extend results from Jacod (2008) [5] to the case of bipower variation, showing under standard assumptions that one obtains a limiting variable, which is in general different from the case of a continuous semimartingale. In a second step a truncated version of bipower variation is constructed, which has a similar asymptotic behaviour as standard bipower variation for a continuous semimartingale and thus provides a feasible central limit theorem for the estimation of the integrated volatility even when the semimartingale exhibits jumps.     

Nonparametric estimation for pure jump Lévy processes based on high frequency data

In this paper, we study nonparametric estimation of the Lévy density for pure jump Lévy processes. We consider n$n$ discrete time observations with step Δ$\Delta$. The asymptotic framework is: n$n$ tends to infinity, Δ=_Δn$\Delta ={\Delta }_n$ tends to zero while n_Δn$n{\Delta }_n$ tends to infinity. First, we use a Fourier approach (“frequency domain”): this allows us to construct an adaptive nonparametric estimator and to provide a bound for the global L²${\mathbb{L}}^2$-risk. Second, we use a direct approach (“time domain”) which allows us to construct an estimator on a given compact interval. We provide a bound for L²${\mathbb{L}}^2$-risk restricted to the compact interval. We discuss rates of convergence and give examples and simulation results for processes fitting in our framework.     

Characterization of the finite variation property for a class of stationary increment infinitely divisible processes

We characterize the finite variation property for stationary increment mixed moving averages driven by infinitely divisible random measures. Such processes include fractional and moving average processes driven by Lévy processes, and also their mixtures. We establish two types of zero–one laws for the finite variation property. We also consider some examples to illustrate our results.     

Convergence to Lévy stable processes under some weak dependence conditions

For a strictly stationary sequence of random vectors in R^d${\mathbb{R}}^d$ we study convergence of partial sum processes to a Lévy stable process in the Skorohod space with _J1$J_1$-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing.     

General change of variable formulas for semimartingales in one and finite dimensions

Summary A general one dimensional change of variables formula is established for continuous semimartingales which extends the famous Meyer-Tanaka formula. The inspiration comes from an application arising in stochastic finance theory. For functions mapping ⁿ to , a general change of variables formula is established for arbitrary semimartingales, where the usualC ² hypothesis is relaxed.Supported in part by NSF grant No. DMS-9103454Supported in part by John D. and Catherine T. MacArthur Foundation award for US-Chile Scientific CooperationSupported in part by FONDECYT, grant 92-0881     

On fractional tempered stable motion

Fractional tempered stable motion (fTSm) is defined and studied. FTSm has the same covariance structure as fractional Brownian motion, while having tails heavier than Gaussian ones but lighter than (non-Gaussian) stable ones. Moreover, in short time it is close to fractional stable Lévy motion, while it is approximately fractional Brownian motion in long time. A series representation of fTSm is derived and used for simulation and to study some of its sample paths properties.     

The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let _ηt${\eta }_t$ be a Poisson point process of intensity t≥1$t\ge 1$ on some state space Y$\mathbb{Y}$ and let f$f$ be a non-negative symmetric function on Y^k${\mathbb{Y}}^k$ for some k≥1$k\ge 1$. Applying f$f$ to all k$k$-tuples of distinct points of _ηt${\eta }_t$ generates a point process _ξt${\xi }_t$ on the positive real half-axis. The scaling limit of _ξt${\xi }_t$ as t$t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m$m$-th smallest point of _ξt${\xi }_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k$k$-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.     

Power variation for Gaussian processes with stationary increments

We develop the asymptotic theory for the realised power variation of the processes X=?•G$X=?•G$, where G$G$ is a Gaussian process with stationary increments. More specifically, under some mild assumptions on the variance function of the increments of G$G$ and certain regularity conditions on the path of the process ?$?$ we prove the convergence in probability for the properly normalised realised power variation. Moreover, under a further assumption on the Hölder index of the path of ?$?$, we show an associated stable central limit theorem. The main tool is a general central limit theorem, due essentially to Hu and Nualart [Y. Hu, D. Nualart, Renormalized self-intersection local time for fractional Brownian motion, Ann. Probab. (33) (2005) 948–983], Nualart and Peccati [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. (33) (2005) 177–193] and Peccati and Tudor [G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, in: M. Emery, M. Ledoux, M. Yor (Eds.), Seminaire de Probabilites XXXVIII, in: Lecture Notes in Math, vol. 1857, Springer-Verlag, Berlin, 2005, pp. 247–262], for sequences of random variables which admit a chaos representation.     

Randomly weighted self-normalized Lévy processes

Let (_Ut,_Vt)$(U_t,V_t)$ be a bivariate Lévy process, where _Vt$V_t$ is a subordinator and _Ut$U_t$ is a Lévy process formed by randomly weighting each jump of _Vt$V_t$ by an independent random variable _Xt$X_t$ having cdf F$F$. We investigate the asymptotic distribution of the self-normalized Lévy process _Ut/_Vt$U_t/V_t$ at 0 and at ∞$\infty$. We show that all subsequential limits of this ratio at 0 (∞$\infty$) are continuous for any nondegenerate F$F$ with finite expectation if and only if _Vt$V_t$ belongs to the centered Feller class at 0 (∞$\infty$). We also characterize when _Ut/_Vt$U_t/V_t$ has a non-degenerate limit distribution at 0 and ∞$\infty$.     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997     

Quasi sure quadratic variation of local times of smooth semimartingales

In this paper, we prove that the process of the quadratic variation of local times of smooth semimartingales can be constructed as the quasi sure limit of the form ∑_Δn(L_t^a_i+1n−L_t^a_in)², where Δ_n=(a_iⁿ,a_i+1ⁿ) is a sequence of subdivisions of [a,b], a_iⁿ=i(b−a)/2ⁿ+a, i=0,1,…,2ⁿ.     

Functional limit theorems for generalized quadratic variations of Gaussian processes

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order quadratic variations, and we illustrate these results with the example of the fractional Brownian sheet (FBS). Second, we do the same study for the process of localized second order quadratic variations, and we apply the results to the multifractional Brownian motion (MBM).