Boundary Values of Harmonic Bloch Functions in Lipschitz Domains: A Martingale Approach

In this paper, the boundary behaviour of a certain class of harmonic functions in Lipschitz domains is studied. It continues the work of N. Makarov on the boudary behaviour of analytic Bloch functions in the unit disk. Certain means associated with the function allow to approximate it by a martingale, and the boudary properties of the function are transferred to the asymptotic behaviour of the martingale.     

Estimating functions for noisy observations of ergodic diffusions

In this article, general estimating functions for ergodic diffusions sampled at high frequency with noisy observations are presented. The theory is formulated in terms of approximate martingale estimating functions based on local means of the observations, and simple conditions are given for rate optimality. The estimation of the diffusion parameter is faster than the estimation of the drift parameter, and the rate of convergence is classical for the drift parameter but not classical for the diffusion parameter. The link with specific minimum contrast estimators is established, as an example.     

Optimal estimation in random coefficient regression models

In linear regression models with random coefficients, the score function usually involves unknown nuisance parameters in the form of weights. Conditioning with respect to the sufficient statistics for the nuisance parameter, when the parameter of interest is held fixed, eliminates the nuisance parameters and is expected to give reasonably good estimating functions. The present paper adopts this approach to the problem of estimation of average slope in random coefficient regression models. Four sampling situations are discussed. Some asymptotic results are also obtained for a model where neither the regressors nor the random regression coefficients replicate. Simulation studies for normal as well as non-normal models show that the performance of the suggested estimating functions is quite satisfactory.     

On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

We compute and then discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of the Ornstein–Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous return process, several measures coincide, some simplifications can be made and the results are more explicit. We illustrate our results with parametric examples used in the literature.     

Estimating functions for jump–diffusions

Asymptotic theory for approximate martingale estimating functions is generalised to diffusions with finite-activity jumps, when the sampling frequency and terminal sampling time go to infinity. Rate-optimality and efficiency are of particular concern. Under mild assumptions, it is shown that estimators of drift, diffusion, and jump parameters are consistent and asymptotically normal, as well as rate-optimal for the drift and jump parameters. Additional conditions are derived, which ensure rate-optimality for the diffusion parameter as well as efficiency for all parameters. The findings indicate a potentially fruitful direction for the further development of estimation for jump–diffusions.     

弱Orlicz鞅空间的弱原子分解及其应用

对3类由凹函数生成的弱Orlicz鞅空间建立了相应的弱原子分解.作为应用,首先给出了这些弱Orlicz鞅空间上次线性算子有界的一个充分条件,并在此基础上证明了一些弱型鞅不等式,然后证明了关于这些弱Orlicz鞅空间的Marcinkiewicz型插值定理.     

强相依数据的函数系数部分线性模型的估计

本文讨论在数据是强相依的情况下函数系数部分线性模型的估计．首先,采用局部线性方法,给出该模型函数项函数的估计;然后,使用两阶段方法给出系数函数的估计．并且讨论了函数项函数估计的渐近正态性,以及系数函数估计的弱相合性和渐近正态性.模拟研究显示,这些估计是较为理想的．     

Distribution of martingales with bounded square functions

We study the terminate distribution of a martingale whose square function is bounded. We obtain sharp estimates for the exponential and p-moments, as well as for the distribution function itself. The proofs are based on the elaboration of the Burkholder method and on the investigation of certain locally concave functions.     

Modular inequalities in martingale Orlicz–Karamata spaces

In this paper, some new martingale inequalities in the framework of Orlicz‐Karamata spaces are provided. More precisely, we establish modular martingale inequalities associated with concave functions and slowly varying functions.     

Joint semiparametric mean-covariance model in longitudinal study

Semiparametric regression models and estimating covariance functions are very useful for longitudinal study. To heed the positive-definiteness constraint, we adopt the modified Cholesky decomposition approach to decompose the covariance structure. Then the covariance structure is fitted by a semiparametric model by imposing parametric within-subject correlation while allowing the nonparametric variation function. We estimate regression functions by using the local linear technique and propose generalized es...     

Partial sums of orthonormal bases preserving positivity—And martingales

We characterize, for finite measure spaces, those orthonormal bases with the following positivity property: if f is a non-negative function, then the partial sums in the expansion of f are non-negative. The bases are necessarily generalized Haar functions and the partial sums are a martingale closed on the right by f.     

Pricing Bounds for Volatility Derivatives via Duality and Least Squares Monte Carlo

Derivatives on the Chicago Board Options Exchange volatility index have gained significant popularity over the last decade. The pricing of volatility derivatives involves evaluating the square root of a conditional expectation which cannot be computed by direct Monte Carlo methods. Least squares Monte Carlo methods can be used, but the sign of the error is difficult to determine. In this paper, we propose a new model-independent technique for computing upper and lower pricing bounds for volatility derivatives. In particular, we first present a general stochastic duality result on payoffs involving convex (or concave) functions. This result also allows us to interpret these contingent claims as a type of chooser options. It is then applied to volatility derivatives along with minor adjustments to handle issues caused by the square root function. The upper bound involves the evaluation of a variance swap, while the lower bound involves estimating a martingale increment corresponding to its hedging portfolio. Both can be achieved simultaneously using a single linear least square regression. Numerical results show that the method works very well for futures, calls and puts under a wide range of parameter choices.     

The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes

For a wide class of local martingales (M _t ) there is a default function, which is not identically zero only when (M _t ) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability of functions of sup _s≤t M _s in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting and last exit times. Received: 6 August 1996 / Revised version: 27 July 1998     

Multiperiodic multifractal martingale measures

A nonnegative 1-periodic multifractal measure on is obtained as infinite random product of harmonics of a 1-periodic function W(t). Such infinite products are statistically self-affine and generalize certain Riesz products with random phases. They are martingale structures, therefore converge. The criterion on W for nondegeneracy is provided. It differs completely from those for other known random measures constructed as martingale limits of multiplicative processes. In particular, it is very sensitive to small changes in W(t). When these infinite products are interpreted in the framework of thermodynamic formalism for random transformations, logW is a potential function when W>0. For regular enough potentials, in case of degeneracy, the natural normalization makes the sequence of measures converge. Moreover, this normalization is neutral for nondegenerate martingales. The multifractal analysis of the limit martingale measure is performed for a class of potential functions having a dense countable set of jump points.     

About the rate function in concentration inequalities for suprema of bounded empirical processes

We provide new deviation inequalities in the large deviations bandwidth for suprema of empirical processes indexed by classes of uniformly bounded functions associated with independent and identically distributed random variables. The improvements we get concern the rate function which is, as expected, the Legendre transform of the suprema of the log-Laplace transform of the pushforward measure by the functions of the considered class (up to an additional corrective term). Our approach is based on a decomposition in martingale together with some comparison inequalities.     

Asymptotic Expansion of M ‐Estimator Over Wiener Space

In this paper we consider an Mestimator defined as a solution of a given estimating function. Sufficient conditions of existence of an Mestimator and its stochastic expansion are presented. In the case where the underlying probability space is a Wiener space and the leading term of the stochastic expansion is a martingale, asymptotic expansions of its distribution function are obtained with the aid of Malliavin calculus. Applications to a stationary ergodic diffusion model are also discussed.     

Hamming cube and martingales

In the present paper, we show that a correctly chosen Legendre transform of the Bellman functions of martingale problems give us the right tool to prove isoperimetric inequalities on Hamming cube independent of the dimension. We illustrate the power of this “dual function approach” by proving certain Poincaré inequalities on Hamming cube and by improving a particular inequality of Beckner on the Hamming cube.     

Semiparametric efficient inferences for lifetime regression model with time-dependent covariates

Through a threshold equation, we propose a time-transformed accelerated failure time (AFT) model with time-dependent covariate history in survival analysis. This model contains a general class of semiparametric lifetime regression models, including AFT with identical time-scale and a wide spectrum of Cox’s hazard regression models and their frailty variants. We first construct the semiparametric efficient statistical inferences on the AFT model with identical time-scale. The theoretical semiparametric Fisher information bound is explicitly derived under right-censored data setting. And the overidentified estimating equation (OEE) approach based on two martingale processes is shown to achieve this semiparametric efficiency bound. Extensions of the semiparametric efficient statistical inferences to the time-transformed AFT versions are also discussed. We also conclude that most log-rank estimating equations would suffer severe information loss primarily caused by wiggling pattern of the baseline hazard function, while the OEE approach can alleviate the damaging effects. A simulated biological life history example is numerically studied.     

Doob's inequality,Burkholder-Gundy inequality and martingale transforms on martingale Morrey spaces

We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob's inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob's inequality on martingale block spaces, we obtain the Davis' decompositions for martingale Morrey spaces.     

Estimating Dependencies from Spatial Averages

Abstract

Modeling of space-time functions can be done using observations in the form of averages of the function over a set of irregularly shaped regions in space-time. Such observations are most common in applications where the data are gathered for administrative, political, geographic, or agricultural regions. The value of such functions can be predicted by first estimating the dependence structure of the underlying stochastic process. Our proposed method for estimating the covariance function from the integrals of a stationary isotropic stochastic process poses the problem as a set of integral equations. To test this proposal we applied it to epidemiological data on the incidence rates of three diseases in the United States between 1980 and 1994. Spatial correlations obtained in this way reasonably described the mechanism by which those diseases spread. We therefore conclude that it is possible to reliably estimate covariance functions from aggregate observations. The estimate of the covariance functions provides valuable insights into the nature of the space-time process—in the epidemiological data it described a possible mechanism by which the diseases spread.