Consistent estimation of covariation under nonsynchronicity

We present a methodology to estimate the covariance of two time series when they are sampled from continuous semimartingales at general stopping times in a nonsynchronous manner. Approximation error bounds being explored, the estimators are shown to be consistent as the size of the sampling intervals tends to zero. The methodology is easy to be implemented with potentially broad applications, especially in financial modeling and analysis involving high-frequency transaction data. The results generalize those recently obtained by obtained by Hayashi and Yoshida (2005, Bernoulli 11(2):359–379)      

Hilbert-值半鞅序列的弱收敛

本文在条件UT下研究了Hilbert-值半鞅序列到连续Hilbert-值半鞅的收敛性,并在弱收敛的条件下研究了形如X^n=∫oa^n(X^n.,s)dY^ns ∫ob^n(X^n.,s)dA^ns,X^no=O,任意n≥1随机微分方程的稳定性,其中Y^n和A^n分别为Hilbert-值半鞅和分量为增过程的Hilbert-值有限变差过程。     

Limit theorems for the pre-averaged Hayashi–Yoshida estimator with random sampling

We will focus on estimating the integrated covariance of two diffusion processes observed in a nonsynchronous manner. The observation data is contaminated by some noise, which possibly depends on the time and the latent diffusion processes, while the sampling times also possibly depend on the observed processes. In a high-frequency setting, we consider a modified version of the pre-averaged Hayashi–Yoshida estimator, and we show that such a kind of estimator has the consistency and the asymptotic mixed normality, and attains the optimal rate of convergence.     

Functional estimation for Lévy measures of semimartingales with Poissonian jumps

We consider semimartingales with jumps that have finite Lévy measures. The purpose of this article is to estimate integral-type functionals of the Lévy measures from discrete observations. We propose two types of estimators: kernel-type and empirical-type estimators, both of which are obtained by direct discretization from asymptotically efficient estimators of the target based on continuous observations. We show the asymptotic efficiency in the asymptotic minimax sense of our estimators as the sample size tends to infinity and the sampling interval tends to zero.     

On existence,uniqueness and convergence of multi-valued stochastic differential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic differential equations driven by continuous semimartingales when the coefficients are stochastically Lipschitz continuous. We also show the convergence results when the random coefficients or the differentials converge.     

On existence,uniqueness and convergence of multi-valued stochastic diferential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic diferential equations driven by continuous semimartingales when the coefcients are stochastically Lipschitz continuous.We also show the convergence results when the random coefcients or the diferentials converge.     

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.     

On properties of strong solutions of stochastic equations with respect to semimartingales

The paper is devoted to strong solutions of stochastic integral equations with respect to semimartingales. We study existence (for non-Lipschitz coefficients) and asymptotic behaviour of strong solutions and obtain also a number of results on weak and square mean convergence of these solutions.     

An estimator for the quadratic covariation of asynchronously observed Itô processes with noise: Asymptotic distribution theory

The article is devoted to the nonparametric estimation of the quadratic covariation of non-synchronously observed Itô processes in an additive microstructure noise model. In a high-frequency setting, we aim at establishing an asymptotic distribution theory for a generalized multiscale estimator including a feasible central limit theorem with optimal convergence rate on convenient regularity assumptions. The inevitably remaining impact of asynchronous deterministic sampling schemes and noise corruption on the asymptotic distribution is precisely elucidated. A case study for various important examples, several generalizations of the model and an algorithm for the implementation warrant the utility of the estimation method in applications.     

Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes

We consider the problem of estimating the covariance of two diffusion-type processes when they are observed only at discrete times in a nonsynchronous manner. In our previous work in 2003, we proposed a new estimator which is free of any ‘synchronization’ processing of the original data and showed that it is consistent for the true covariance of the processes as the observation interval shrinks to zero; Hayashi and Yoshida (Bernoulli, 11, 359–379, 2005). This paper is its sequel. Specifically, it establishes asymptotic normality of the estimator in a general nonsynchronous sampling scheme.     

Extended convergence to continuous in probability processes with independent increments

Summary A special (extended) kind of convergence in distribution of processes with filtration is considered. Recent theorems on the functional convergence of semimartingales are improved by showing that their assumptions imply the extended convergence of semimartingales to continuous in probability processes with independent increments.     

Rates of convergence of diffusions with drifted Brownian potentials

We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.

     

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.     

A strong uniform time for random transpositions

A random permutation ofN items generated by a sequence ofK random transpositions is considered. The method of strong uniform times is used to give an upper bound on the variation distance between the distributions of the random permutation generated and a uniformly distributed permutation. The strong uniform time is also used to find the asymptotic distribution of the number of fixed points of the generated permutation. This is used to give a lower bound on the same variation distance. Together these bounds give a striking demonstration of the threshold phenomenon in the convergence of rapidly mixing Markov chains to stationarity.     

Semimartingale stochastic approximation procedure and recursive estimation

The semimartingale stochastic approximation procedure, precisely, the Robbins-Monro type SDE, is introduced, which naturally includes both generalized stochastic approximation algorithms with martingale noises and recursive parameter estimation procedures for statistical models associated with semimartingales. General results concerning the asymptotic behavior of the solution are presented. In particular, the conditions ensuring the convergence, the rate of convergence, and the asymptotic expansion are established. The results concerning the Polyak weighted averaging procedure are also presented. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 45, Martingale Theory and Its Application, 2007.     

Empirical distributions in marked point processes

We study the asymptotic behaviour of the empirical distribution function derived from a stationary marked point process when a convex sampling window is expanding without bounds in all directions. We consider a random field model which assumes that the marks and the points are independent and admits dependencies between the marks. The main result is the weak convergence of the empirical process under strong mixing conditions on both independent components of the model. Applying an approximation principle weak convergence can be also shown for appropriately weighted empirical process defined from a stationary d-dimensional germ-grain process with dependent grains.     

On Statistical Inference Based on Record Values

We develop methodology for conducting inference based on record values and record times derived from a sequence of independent and identically distributed random variables. The advantage of using information about record times as well as record values is stressed. This point is a subtle one, since if the sampling distribution F is continuous then there is no information at all about F in the record times alone; the joint distribution of any number of them does not depend on F. However, the record times and record values jointly contain considerably more information about F than do the record values alone. Indeed, in the case of a distribution with regularly varying tails, the rate of convergence of the exponent of regular variation is two orders of magnitude faster if information about record times is included. Optimal estimators and convergence rates are derived under simple, specific models, and shown to be surprisingly robust against significant departures from those models. However, even under our special models the estimators have irregular properties, including an undefined information matrix. To some extent these difficulties may be alleviated by conditioning and by considering the relationship between maximum likelihood and maximum probability estimators.     

A Differentiation Theory for Itô's Calculus

Abstract

A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? From Itô's definition of his integral, such a derivative must be based on the quadratic variation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications, some of which we address in an upcoming article.     

Strong approximations of semimartingales by processes with independent increments

Summary Strong approximation theorems for continuous time semimartingales are obtained by combining some techniques of the general theory of stochastic processes with some of the direct approximation of dependent random variables by independent ones. Continuous processes with independent increments whose variance functions increase polynomially or exponentially are considered as approximating processes. The basic assumptions of the main results only contain rates of convergence for certain probabilities. In particular, moment assumptions are not required. Some almost sure invariance principles for partial sum processes with nonlinear growth of variance and for functionals of Markov processes are derived by applying the main results.     

An Itō Formula in the Space of Tempered Distributions

We extend the Itō formula (Rajeev in From Tanaka’s formula to Ito’s formula: distributions, tensor products and local times, Springer, Berlin, 2001, Theorem 2.3) for semimartingales with paths that are right continuous and have left limits. We also comment on the local time process of such semimartingales. We apply the Itō formula to Lévy processes to obtain existence of solutions to certain classes of stochastic differential equations in the Hermite–Sobolev spaces.