Multiparameter and set-indexed stochastic processes

多参数和集合索引随机过程

• 批准号: RGPIN-2014-05613
• 负责人: Ivanoff, BarbaraGail
• 金额: $ 1.31万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=654351
• 关键词: Multiparameter; set; indexed; stochastic; processes; Multiparameter; set; indexed; stochastic; processes

Random processes indexed by a multidimensional time parameter, or more generally by a class of sets, may be used to model many stochastic phenomena and have been the focus of much research activity in recent years. However, the passage from a totally ordered index set (time) to one that is partially ordered (multidimensional time or a class of sets) introduces many challenges. In particular, describing the dynamical properties of the process (i.e. how the process evolves) becomes far more complex. Concepts such as the martingale or renewal property that depend on the ideas of "past" and "future" must be extended to this more general framework. This is particularly important since martingale methods provide elegant and powerful nonparametric methods for point process inference, survival analysis, state estimation, change point problems, and easily incorporate censored data.**In this research, we will be further developing fundamental properties of multi-parameter martingales, and then we will exploit this theory to develop new statistical techniques for the analysis of various examples of stochastic processes indexed by a multidimensional time parameter. Such processes include multi-parameter time series, renewal processes and empirical processes. Furthermore, multi-parameter martingale methods can be applied to survival analysis of data that has been censored by a random set (i.e. the process is obscured outside of a random set). We will also examine the converse problem, in which the stochastic process is totally observed, but its behaviour can change on a random set that cannot be observed. In this case, we will develop techniques to detect the existence of the so-called change set.**These techniques will have applications in diverse areas, including economics, finance, geography, geology, biology and the environment.

由多维时间参数索引的随机过程，或更普遍地由一类集合用于对许多随机现象进行建模，并且近年来一直是许多研究活动的重点。但是，从完全有序的索引集（时间）到部分有序的索引集（多维时间或一类集合）的段落引入了许多挑战。特别是，描述过程的动力学特性（即过程如何发展）变得更加复杂。依赖于“过去”和“未来”的思想的概念，例如Martingale或更新属性，必须扩展到这个更一般的框架。这一点尤其重要，因为马丁加尔方法为要点过程推理，生存分析，州估计，变更点问题提供了优雅而有力的非参数方法，并且很容易纳入审查的数据。 范围。这样的过程包括多参数时间序列，更新过程和经验过程。此外，可以将多参数martingale方法应用于对随机集合进行审查的数据的存活分析（即，在随机集合之外掩盖了该过程）。我们还将检查相反的问题，其中完全观察到随机过程，但是它的行为可以在无法观察到的随机集中发生变化。在这种情况下，我们将开发技术来检测所谓的变更集的存在。