鞅

Based on a multivariate extension of the constrained locally polynomial estimator of Ait-Sahalia and Duarte (2003), we provide one of the first nonparametric estimates of probability densities of LIBOR rates under forward martingale measures and state-price densities implicit in interest rate cap prices.

根据局部性 Ait-Sahalia 与 Duarte (2003)多项式估计的多元扩展，我们提供了基于远期鞅方法首个伦敦银行同业拆借利率非参数估计的概率密度，以及隐性利率上限价格中的国家价格密度。

To make this market a complete one, we introduce the forward rate, which is driven by this Brownian motion, as a new risk factor, and under HJM model we have the new asset - zero-coupon bond. Then we can get the price of stock index futures by the martingale method.

为此引入受这个Brown运动影响的远期利率作为风险因子，在HJM模型框架下可化作新的风险资产——零息债券，得到虚拟的完全化市场后利用鞅方法解出股指期货价格表达式。

There has been various betting strategies and tactics that hope to overcome this odd to beat the casino, such as Martingale betting strategy and Fibonacci sequence, although none has proved to be to a complete success.

有已各投注的战略和战术，希望克服这种奇怪的击败赌场，如鞅投注策略和Fibonacci序列，虽然没有已被证明是一个取得了圆满成功。

There has been various betting strategies and tactics that hope to overcome this odd to beat the casino, such as Martingale betting strategy and Fibonacci sequence, although none has proved to be to a complete success.

已有各种博彩战略和战术，希望克服这种奇怪击败赌场，如鞅博彩战略和Fibonacci数列，但没有已被证明是完全成功。

Formula and its conclusions,which are the basis of our computations in this thesis.

通过选取不同的计价单位，我们可以得到不同的鞅测度。

First of all, a class of strong limit theorems which extend the conclusions in [29] are proved by constructing two nonnegative martingales.

首先，通过构造两个非负鞅证明了一类强极限定理，从而推广了[29]中的结论。

Fourier analysis of distributions, central limit problem and infinitely divisible laws, conditional expectations, martingales.

分布的Fourier分析，中心极限问题和无限可分法则，条件期望，鞅。

It differs from the scalar-valued case that the existence of the atomic decompositions of Banach-space-valued martingales is closely connected with the geometrical properties of the Banach spaces.

与标量值不同的是，在向量值情形下，这些鞅空间的某些类型的原子分解的存在性和它们之间的相互关系都与值空间的几何性质有紧密联系。

Prof. Long-Ruilin mentioned that if we want to study weighted inequalities for martingales, such as the maximal inequalities, the inequalities for the square operator and so on, the condition imposed on weights are mainly A〓 and S, but we don't know whether this condition is superfluous in all problems we consider. Since the condition can not be weakened for the known operators, now we consider a new operator to discuss its L〓-boundedness, then another problem arises, under what condition can we get it?

龙瑞麟先生曾经提到要讨论关于可测的鞅的极大算子、均方根算子以及条件均方根算子的L〓有界性，或它们之间的〓有界性时，所设的主要条件是A〓与条件S，有了这两个条件在讨论加权问题时基本上没有什么困难，但是要想用更弱的条件来代替它们，即使在古典情况下（此时S条件是自然满足的）也恐怕还不到时候。

Course topics will be selected from: the general theory of processes, sample function properties, weak convergence, Brownian motion, diffusions, Levy processes, Markov processes, martingales, Gaussian processes and further topics.

课程论题将从以下选择：过程的一般理论，样本函数的特性，弱收敛，布朗运动，扩散，Levy过程，马氏过程，鞅，高斯过程及更深的论题。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。