Another view on martingale central limit theorems

Based on the martingale version of the Skorokhod embedding Heyde and Brown (1970) established a bound on the rate of convergence in the central limit theorem (CLT) for discrete time martingales having finite moments of order 2+2δ with 0<δ1. An extension for all δ>0 was proved in Haeusler (1988). This paper presents a rather quick access based solely on truncation, optional stopping, and prolongation techniques for martingale difference arrays to obtain other upper bounds for sup

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(φbeing the standard normal d.f.) yielding weak sufficient conditions for the asymptotic normality of . It is shown that our approach also yields two types of martingale central limit theorems with random norming.     

Stochastic regression model with heteroscedastic disturbance

This paper discusses some properties of stochastic regression model with continuous form of heteroscedastic disturbance. The strong consistency and asymptotic normality of a generalized weighted least squares estimate will be investigated under certain conditions on the stochastic regressors and errors. More, the linear hypothesis testing problem also be discussed and an example to be demonstrated to reestablish the results of Cheng and Chang (1990, Tech. Report, National Tsing Hua University).Supported by the National Science Council Grant No. 810208M763 at National Tsing Hua University.     

停止变换的不变性

利用条件期望的表达式给出了两指标过程在停线处的停止定义,研究了停止变换下的若干不变性．给出了这种停止意义下的局部平方可积强鞅的定义,进一步研究了局部平方可积强鞅二次变差的存在性及其停止性质，得到了重要的Burkholder-Davis-Gundy型不等式及平方可积强鞅的一个充要条件。     

Estimation of IIR Systems with Binary-Valued Observations*

Estimation and control problems with binary-valued observations exist widely in practical systems. However, most of the related works are devoted to finite impulse response (FIR for short) systems, and the theoretical problem of infinite impulse response (IIR for short) systems has been less explored. To study the estimation problems of IIR systems with binary-valued observations, the authors introduce a projected recursive estimation algorithm and analyse its global convergence properties, by using the stochastic Lyapunov function methods and the limit theory on double array martingales. It is shown that the estimation algorithm has similar convergence results as those for FIR systems under a weakest possible non-persistent excitation condition. Moreover, the upper bound for the accumulated regret of adaptive prediction is also established without resorting to any excitation condition.     

On stationary Markov processes with polynomial conditional moments

We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say m is a polynomial of degree at most m. We show that then under some additional, natural technical assumption there exists a family of orthogonal polynomial martingales. More precisely we show that such a family of processes is completely characterized by the sequence {(α_n, p_n)}_{n ? 0} where α′_ns are some positive reals while p′_ns are some monic orthogonal polynomials. Bakry and Mazet (Séminaire de Probabilit?s, vol. 37, 2003) showed that under some additional mild technical conditions each such sequence generates some stationary Markov process with polynomial regression.

We single out two important subclasses of the considered class of Markov processes. The class of harnesses that we characterize completely. The second one constitutes of the processes that have independent regression property and are stationary. Processes with independent regression property so to say generalize ordinary Ornstein–Uhlenbeck (OU) processes or can also be understood as time scale transformations of Lévy processes. We list several properties of these processes. In particular we show that if these process are time scale transforms of Lévy processes then they are not stationary unless we deal with classical OU process. Conversely, time scale transformations of stationary processes with independent regression property are not Lévy unless we deal with classical OU process.     

由Levy过程驱动的倒向随机微分方程在随机单调条件下解的存在唯一性

本文讨论了如下的由Levy过程驱动的倒向随机微分方程适应解的存在唯一性■其中W_s是一Wiener过程,H_s为由Levy过程构成Teugels鞅.我们通过构造函数逼近序列的方法证明了,在漂移系数f关于Y满足随机单调,f关于Z和U满足随机Lipschitz条件下,方程存在唯一适应解.     

Duality and martingales: a stochastic programming perspective on contingent claims

The hedging of contingent claims in the discrete time, discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program. Application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics, namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale. The model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims. However, we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options. An extension of the model to incorporate pre-existing liabilities and endowments reveals the reasons why buyers and sellers trade in options. The model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets. Received: June 5, 2000 / Accepted: July 12, 2001?Published online December 6, 2001     

On the characterization of certain similarly ordered super-additive functionals

Functionals which behave (sub-, super-) additively on similarly ordered functions occur quite naturally in many contexts. In the present paper we characterize (super-) additive functionals which are defined on a family of functions with the Stone-property in terms of their naturally adjoined dyadic martingales. As corollaries we obtain essential generalizations of integral representations as derived by Schmeidler (1986) and discussed in a recent monograph of Denneberg (1994).

     

A note on confidence sequences in multiparameter exponential families

In a paper by Lai [(1976) Ann. Statist.4] a construction of sequences of confidence intervals for one-parameter exponential families is given, where the intersection of these intervals contains the true parameter value θ₀ with a prescribed level of confidence and the respective interval bounds tend a.s. to θ₀. In this paper an analogous result for multiparameter exponential families is proved. More presicely, sequences of convex confidence sets are obtained which shrink to the actual parameter value.     

Some analysis of a stochastic logistic growth model

We derive several new results on a well-known stochastic logistic equation. For the martingale case, we compute the distribution of the solution, mean passage times, and the distribution of hitting times, all in closed form. For the case of constant coefficients, we also find mean passage times and for the general equation we give the weak solution expressed in terms of stochastic quadratures. We also show how these quadratures may be considerably simplified using the results for the martingale case. As it turns out, the martingale case has a particularly elegant weak solution, and to a large degree its structure carries over to the general case.