Multiparameter and set-indexed stochastic processes

由多维时间参数或更一般地由一类集合标引的随机过程可用于对许多随机现象进行建模，并已成为近年来许多研究活动的焦点。然而，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)带来了许多挑战。特别是，描述过程的动态属性(即过程如何演变)变得复杂得多。依赖于“过去”和“未来”概念的一些概念，如鞅或更新性质，必须扩展到这个更一般的框架。这一点尤其重要，因为鞅方法为点过程推断、生存分析、状态估计、变点问题提供了优秀而强大的非参数方法，并且很容易结合截尾数据。**在本研究中，我们将进一步发展多参数鞅的基本性质，然后我们将利用这一理论来开发新的统计技术来分析以多维时间参数为指标的随机过程的各种实例。这些过程包括多参数时间序列、更新过程和经验过程。此外，多参数鞅方法可以应用于被随机集删失(即过程在随机集之外被遮蔽)的数据的生存分析。我们还将研究逆问题，在这个问题中，随机过程完全可以观察到，但它的行为可以在一个无法观察到的随机集上改变。在这种情况下，我们将开发技术来检测所谓的变化集的存在。**这些技术将在不同的领域得到应用，包括经济、金融、地理、地质、生物和环境。

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.