Estimation of parameters of SDE driven by fractional Brownian motion with polynomial drift

ABSTRACT

Strongly consistent and asymptotically normal estimators of the Hurst index and volatility parameters of solutions of stochastic differential equations with polynomial drift are proposed. The estimators are based on discrete observations of the underlying processes.     

Bounds for the expected value of the stochastic Divisia's price index

In this article, we estimate bounds for the expected value of the stochastic Divisia's price index, that is, we assume that prices and quantities of the given commodities are stochastic processes with continuous time. We consider some special case of the stochastic model in which prices and quantities are described by the geometric Brownian motion. It is shown that the precision of this estimation depends rather on the volatility of prices than quantities volatilities.     

Some Nonparametric Asymptotic Results for a Class of Stochastic Processes

This article provides a solution of a generalized eigenvalue problem for integrated processes of order 2 in a nonparametric framework. Our analysis focuses on a pair of random matrices related to such integrated process. The matrices are constructed considering some weight functions. Under asymptotic conditions on such weights, convergence results in distribution are obtained and the generalized eigenvalue problem is solved. Differential equations and stochastic calculus theory are used.     

Analysis of Stochastic Online Bin Packing Processes

Abstract

A stochastic online version of the classical bin packing problem, where a bin corresponds to the capacity of a resource allocated among streams of requests at discrete time units, is a fundamental problem that arises in a wide variety of application areas including bandwidth allocation in networks, memory management in computers, and message transmission in slotted network channels. We derive a mathematical analysis of the corresponding multi-dimensional stochastic process, potentially infinite in each dimension, under a general class of scheduling policies based on a combination of a Lyapunov function technique and matrix-analytic methods. Our analysis yields stability conditions and stationary distributions for this stochastic bin packing process under general probability distributions. We further provide some algorithmic techniques for the numerical computation of these measures.     

Stochastic comparison of processes generated by random interruptions of monotone functions and related results

Many stochastic processes considered in applied probability models, and, in particular, in reliability theory, are processes of the following form: Shocks occur according to some point process, and each shock causes the process to have a random jump. Between shocks the process increases or decreases in some deterministic fashion. In this paper we study processes for which the rate of increase or decrease between shocks depends only on the height of the process. For such processes we find conditions under which the processes can be stochastically compared. We also study hybrid processes in which periods of increase and periods of decrease alternate. A further result yields a stochastic comparison of processes that start with a random jump, rather than processes in which there is at the beginning some random delay time before the first jump.Supported by NSF Grant DMS 9303891.     

Estimation of error probabilities in stochastic dominance

The process of using data to infer the existence of stochastic dominance is subject to sampling error. Kroll and Levy (1980), among others, have presented simulation results for several normal and lognormal distributions which show high error probabilities for a wide range of parameter values. This paper continues this line of research and uses simulation to estimate error probabilities. Distributions considered are a pair of normals and a pair of lognormals. Analysis of these distributions is made computationally feasible through theoretical results which reduce the number of parameters of the pair of distributions from four to two.     

Stochastic simulation of sequential game-theory voting models

We discuss the use of stochastic simulation as a tool to learn about optimal behavior and Nash equilibria of a sequential voting model proposed by Osborne, related to Duverger’s law. We introduce a graphical Java applet, which implements such simulations and investigates its properties. We show that in an appropriate setup, the applet is guaranteed to eventually find behavior, which is within ε of being optimal.     

Road trafficking description and short term travel time forecasting,with a classification method

The purpose of this work is, on the one hand, to study how to forecast road trafficking on highway networks and, on the other hand, to describe future traffic events. Here, road trafficking is measured by vehicle velocities. The authors propose two methodologies. The first is based on an empirical classification method, and the second on a probability mixture model. They use an SAEM‐type algorithm (a stochastic approximation of the EM algorithm) to select the densities of the mixture model. Then, they test the validity of their methodologies by forecasting short term travel times.     

Statistical games against a prophet-proof of a minimax conjecture

“Prophet theory” quantifies the price a statistician has to pay for his lack of information in stochastic sequences. In a recent paper, Schmitz (1991) gave a game-theoretical interpretation of this situation and he formulated in particular a minimax conjecture for the difference case. In this note we prove that conjecture and, moreover, present minimax ran domized stopping times (minimax procedures for the statistician).     

突发事件对国际石油期货价格波动的时间记忆性分析——基于PPM模型和Hurst指数分析

以国际原油期货价格波动日度数据为样本,通过PPM突变点识别模型,从高频时间序列中出现的众多"跳跃点"中甄别出能够改变时间序列波动趋势的突变点,对相邻两突变点之间样本进行Hurst指数分析,研究突发事件对国际原油期货价格波动的时间记忆性。研究表明,时间序列数据中存在多个"跳跃点",大多数跳跃点并未改变数据波动趋势,少数突变点不但改变数据波动趋势,国际原油期货价格波动受突发事件的影响改变其原有的运动轨迹。此项研究为后续学者提供了一种研究"事件冲击"和"数据时间记忆"的新方法。     

Nonparametric Entropy-Based Tests of Independence Between Stochastic Processes

This article develops nonparametric tests of independence between two stochastic processes satisfying β-mixing conditions. The testing strategy boils down to gauging the closeness between the joint and the product of the marginal stationary densities. For that purpose, we take advantage of a generalized entropic measure so as to build a whole family of nonparametric tests of independence. We derive asymptotic normality and local power using the functional delta method for kernels. As a corollary, we also develop a class of entropy-based tests for serial independence. The latter are nuisance parameter free, and hence also qualify for dynamic misspecification analyses. We then investigate the finite-sample properties of our serial independence tests through Monte Carlo simulations. They perform quite well, entailing more power against some nonlinear AR alternatives than two popular nonparametric serial-independence tests.     

Fragility index of block tailed vectors

Financial crises are a recurrent phenomenon with important effects on the real economy. The financial system is inherently fragile and it is therefore of great importance to be able to measure and characterize its systemic stability. Multivariate extreme value theory provide us such a framework through the fragility index ( 11, 7 and 8). Here we generalize this concept and contribute to the modeling of the stability of a stochastic system divided into blocks. We will find several relations with well-known tail dependence measures in the literature, which will provide us immediate estimators. We end with an application to financial data.     

Generalized preliminary test stochastic restricted estimator in the linear regression model

ABSTRACT

In this paper, we propose three generalized estimators, namely, generalized unrestricted estimator (GURE), generalized stochastic restricted estimator (GSRE), and generalized preliminary test stochastic restricted estimator (GPTSRE). The GURE can be used to represent the ridge estimator, almost unbiased ridge estimator (AURE), Liu estimator, and almost unbiased Liu estimator. When stochastic restrictions are available in addition to the sample information, the GSRE can be used to represent stochastic mixed ridge estimator, stochastic restricted Liu estimator, stochastic restricted almost unbiased ridge estimator, and stochastic restricted almost unbiased Liu estimator. The GPTSRE can be used to represent the preliminary test estimators based on mixed estimator. Using the GPTSRE, the properties of three other preliminary test estimators, namely preliminary test stochastic mixed ridge estimator, preliminary test stochastic restricted almost unbiased Liu estimator, and preliminary test stochastic restricted almost unbiased ridge estimator can also be discussed. The mean square error matrix criterion is used to obtain the superiority conditions to compare the estimators based on GPTSRE with some biased estimators for the two cases for which the stochastic restrictions are correct, and are not correct. Finally, a numerical example and a Monte Carlo simulation study are done to illustrate the theoretical findings of the proposed estimators.     

Matrix‐Exponential Distributions: Calculus and Interpretations via Flows

By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace–Stieltjes transforms on [0,∞). That approach includes and generalizes the formalism of PH-distributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allows for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP).     

非对称随机波动模型在中国股市的应用

为了探测随机波动模型的非对称特征,修改传统的随机波动模型建立非对称的随机波动模型,采用基于马尔可夫链蒙特卡洛(MCMC)模拟的贝叶斯分析对模型进行参数估计。对中国深圳、上海股市波动进行实证研究发现,非对称随机波动模型能较好地探测波动存在的非对称波动。与GJR-GARCH模型相比,非对称随机波动模型预测效果更好。     

基于广义双曲线SV模型的极值风险度量研究

在金融风险的度量中,拟合分布的选取直接影响到风险度量的精度问题。针对金融收益序列的动态变化,在SV模型中引入广义双曲线学生偏t分布(SV-GHSKt)拟合金融收益序列的尖峰厚尾、不对称以及杠杆效应等特征,通过马尔科夫蒙特卡洛模拟的方法将收益率序列转化为标准残差序列,然后用极值理论的POT模型拟合标准残差序列尾部分布,进而建立一种新的金融风险度量模型———基于SV-GHSKt-POT的动态VaR模型。用该模型对上证综合指数做实证研究,结果表明,SV-GHSKt-POT的动态VaR模型能很好地模拟金融收益序列的尖峰厚尾性、波动集聚性及杠杆效应,并且能够合理有效地提高风险测度的精度,尤其在高的置信水平下表现更好。     

Derivative Pricing With Wishart Multivariate Stochastic Volatility

This paper deals with the pricing of derivatives written on several underlying assets or factors satisfying a multivariate model with Wishart stochastic volatility matrix. This multivariate stochastic volatility model leads to a closed-form solution for the conditional Laplace transform, and quasi-explicit solutions for derivative prices written on more than one asset or underlying factor. Two examples are presented: (i) a multiasset extension of the stochastic volatility model introduced by Heston (1993), and (ii) a model for credit risk analysis that extends the model of Merton (1974) to a framework with stochastic firm liability, stochastic volatility, and several firms. A bivariate version of the stochastic volatility model is estimated using stock prices and moment conditions derived from the joint unconditional Laplace transform of the stock returns.     

On two conjectures about perturbations of the stochastic growth rate

The stochastic growth rate describes long-run growth of a population that lives in a fluctuating environment. Perturbation analysis of the stochastic growth rate provides crucial information for population managers, ecologists and evolutionary biologists. This analysis quantifies the response of the stochastic growth rate to changes in demographic parameters. A form of this analysis deals with changes that only occur in some environmental states. Caswell put forth two conjectures about environment-specific perturbations of the stochastic growth rate. The conjectures link the stationary distribution of the stochastic environmental process with the magnitude of some environment-specific perturbations. This note disproves one conjecture and proves the other.     

A generalization of the Solis-Wets method

In this paper we focus on the application of global stochastic optimization methods to extremum estimators. We propose a general stochastic method—the master method—which includes several stochastic optimization algorithms as a particular case. The proposed method is sufficiently general to include the Solis-Wets method, the improving hit-and-run algorithm, and a stochastic version of the zigzag algorithm. A matrix formulation of the master method is presented and some specific results are given for the stochastic zigzag algorithm. Convergence of the proposed method is established under a mild set of conditions, and a simple regression model is used to illustrate the method.