Mathematical Sciences: Mathematical Logic and Applications

这个项目的一个主要目标是利用非标准分析给出一个一般的元定理，它将连续时间随机过程的存在性问题归结为关于离散时间过程的相应陈述。即将发表的题为“从离散时间到连续时间”的文件朝着这一目标迈出了一步。本文引入了强迫的概念，并证明了一大类关于连续时间随机过程的陈述成立的充要条件是被一列离散随机过程强迫。这个强迫定理可以用来证明各种存在定理。例如，该方法证明了许多随机微分方程解的存在性。离散时间语句通常很容易证明，而强迫定理完成了从那里到连续时间语句所需的大部分工作。从数理逻辑的角度来看，强迫定理目前的表述方式是自然的，但应用起来往往比人们所希望的要困难得多。这样做的一个原因是，人们必须首先证明要证明的陈述是可以用“可提升函数”来表达的，“可提升函数”是使用非标准分析以技术方式定义的。研究人员正在研究一种更普遍、更方便的强迫定理，它将在证明中使用非标准概念，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程序来测试该方法的范围，以解决概率论中的特定问题。令人惊讶的是，数理逻辑中的深层方法可以用来在概率论中获得替代的、往往更简单的证明。然而，在多年的时间里，这位调查人员通过实例非常成功地证明了这一点。

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.