鞅

Under the assumptiom that the price procee of risky asset is assumed to be a semimartingale under original probability measure, we prove existence and uniqueness of the locally risk-minimizing strategy.

我们在风险资产的价格过程在原概率测度下为半鞅的假设下，证明了局部风险最小对冲策略的存在性和唯一性。

In Chapter 3 Firstly, the exponential stability in mean square and almost sure exponential stability for neutral stochstic neural networks with time-varying delay can be discussed by using the linear matrix inequality and the semimartingale convergence theorem; And our sufficient conditions are less conserative; Secondly, by utilizing the fixed point theorem, some sufficient and necessary conditions ensuring the asymp- totic stability in mean square for neutral stochastic neural networks with time-varying delays can be given.

第三章首先，利用线性矩阵不等式方法和半鞅收敛定理，考虑了一类具变时滞中立型随机神经网络的二阶矩指数稳定性和几乎必然指数稳定性，所得到的条件保守性要好；其次，采用不动点方法，给出了具变时滞中立型随机神经网络的二阶矩渐近稳定性的充要条件。

Motivated by the newψfunctional stability,we can establish some new kinds of stabilities which are different from the exponential stability.With the help of semimartingale convergence theorem,stochastic calculous,moment inequality,Burkholder-Davis-Gundy inequality,Gronwall inequality etc.,we find that the product ofψfunction and the solutions of neutral equations will approach a nonempty set if some specific conditions are satisfied.

我们应用半鞅收敛定理、随机微积分的相关知识以及矩不等式、Burkholder-Davis-Gundy不等式和Gronwall不等式等技巧，得到中立型随机泛函微分方程的解与特定的"ψ函数"的乘积在一定的条件下将趋向某非空集合这一结论，并建立了该中立型方程的随机LaSalle型定理。

Neutral equation; stochastic functional differential equation; stochastic differential equation with delay; infinite delay; asymptotic propertie; semimartingale convergence theorem; Markovian switching; moment stability; moment boundedness

基础科学，数学，概率论、数理统计中立型方程；随机泛函微分方程；随机延迟微分方程；无限时滞；渐近性质；半鞅收敛定理； Markov调制；矩稳定性；矩有界性

In the fourth chapter,we discuss theapplication of the atomic decomposition theory to the vector-valued harmonicanalysis.We give the Hardy inequalities in the vector-valued martingale space,we also study the Mikhlin multiplier operator and singular integral operator in thevector-valued Hardy space.

第四章我们主要研究原子分解理论在向量值调和分析中的应用，建立了向量值鞅空间上的Hardy不等式，向量值Hardy空间上的Mikhlin乘子算子及一类奇异积分算子的有界性。

Real options theory, which combines corporate finance, investment decisions and management science, deeply applys mathematical tools, such as stochastic calculus, optimal stopping times, martingale, stochastic differential equations, game theory, etc. Real options analysis overcomes the drawbacks of traditional methods, and provides a new method of investment decisions, financing policies and risk management for the corporate.

实物期权方法深入应用随机分析、最优停时、鞅、随机微分方程、博弈论等数学工具，并与金融学、投资决策和公司管理等学科相互结合，克服了传统方法的局限，为企业提供了一套投资财务战略管理和风险管理的新方法。

Real options theory, which combines corporate finance, investment decisions and management science, deeply applys mathematical tools, such as stochastic calculus, optimal stopping times, martingale, stochastic differential equations, game theory, etc.

实物期权方法深入应用随机分析、最优停时、鞅、随机微分方程、博弈论等数学工具，并与金融学、投资决策和公司管理等学科相互结合，克服了传统方法的局限，为企业提供了一套投资财务战略管理和风险管理的新方法。

Martingales, optimal stopping, Wald's lemma, age-dependent branching processes, stochastic integration, Ito's lemma.

随机过程：鞅，最优停步，Wald引理，依赖年龄的分支过程，随机积分， Ito引理。滑铁卢大学本科，研究生

Therefore, it's natural to estimate those values of operators such as 〓, log mf and so on in order to investigate the weighted inequalities for the minimal operator We discuss them under two conditions respectively: In one-weight case, we establish the relative weighted inequalities under the condition of A〓-weight and A〓-weight respectively, and the equivalent condition has been found between the strong -type weighted inequality and the property of W〓-weight; while in the case of two-weight condition, we characterize the equivalent relations between the weak type weighted inequality and the property of W〓-weight, the strong -type weighted inequality and the property of W〓-weight.

我们分两种情况分别加以讨论：单权情况下，我们分别刻画了A∞权和A〓权条件下关于〓，log mf等的强型加权不等式，并证明了关于鞅的极小算子的强型加权不等式与对应的W〓权性质之间的某些等价关系；在双权情况下，我们刻画了关于极小算子的弱型加权不等式与W〓权以及强型加权不等式与W〓权之间的等价关系。

Piecewise deterministic Markov processes have important application in risk theory, especially the theory of martingal on the basis of PDMP has offered the method to stylize for studying risk theory.

逐段决定马尔可夫过程在风险理论中具有重要的应用，特别是基于逐段决定马尔可夫过程的鞅理论为研究风险理论提供了程式化的方法。

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。