Mathematical Sciences: Mathematical Logic and Applications

数学科学：数学逻辑及其应用

• 批准号: 9024411
• 负责人: H. Jerome Keisler
• 金额: $ 12.87万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-06-15 至 1995-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9024411&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Logic; Applications

A major objective of this project is to use nonstandard analysis to give a general metatheorem which reduces existence problems for continuous-time stochastic processes to corresponding statements about discrete-time processes. A step towards the objective was taken in the forthcoming paper, "From Discrete to Continuous Time". In that paper, a concept of forcing was introduced, and it was proved that a wide class of statements about continuous-time stochastic processes are true if and only if forced by a sequence of discrete-time stochastic processes. This forcing theorem can be used to prove a variety of existence theorems. For example, the method gives proofs of the existence of solutions of many stochastic differential equations. The discrete-time statement is often easy to prove, and the forcing theorem does most of the work required to get from there to the continuous-time statement. The forcing theorem is presently formulated in a way which is natural from the point of view of mathematical logic but is often more difficult to apply than one would hope. One reason for this is that one must first show that the statement to be proved is expressible in terms of "liftable functions", which are defined in a technical way using nonstandard analysis. The investigator is working toward a more general and more convenient forcing theorem which would use nonstandard notions in its proof but not in its application. To complement this effort, the scope of the method will be tested by developing a wider variety of applications to particular problems in probability theory. It is surprising that deep methods from mathematical logic can be used to obtain alternative and often simpler proofs in the theory of probability. However, the investigator has been very successful in demonstrating this by example over a period of many years.

该项目的主要目标是使用非标准 分析给出了一个一般的元定理，减少了存在 连续时间随机过程问题 关于离散时间过程的相应陈述。 在即将到来的 《从离散时间到连续时间》 在这篇文章中，A 引入了强迫的概念，证明了一个广泛的 一类关于连续时间随机过程的陈述 是真的，当且仅当由离散时间序列强制 随机过程 这个强迫定理可以用来证明 各种存在定理。 例如，该方法给出 多个随机变量解的存在性的证明 微分方程 离散时间语句通常是 很容易证明，强迫定理做了大部分工作 从那里到连续时间语句。 强迫定理目前是以一种 从数理逻辑的角度来看是自然的， 往往比人们希望的更难申请。 一个原因 因为这是一个必须首先表明，声明是 证明是可表达的“提升功能”，这是 使用非标准分析的技术方法定义。 的 调查人员正在努力建立一个更普遍、更方便的 在证明中使用非标准概念的强迫定理 而不是在其应用中。 为了补充这一努力， 将通过开发更广泛的 应用于概率论中的特殊问题。 令人惊讶的是，来自数理逻辑的深层方法 可以用来获得替代和往往更简单的证明， 概率论 然而，调查人员一直非常 成功地证明了这一点，在一段时间内，许多 年