Statistics and dynamics of conditioned random systems

这项研究旨在扩大我们对随机动力系统的理解，即其潜在的进化规律是由噪声驱动的系统。对这些系统的透彻理解是至关重要的，因为金融、生态或气候科学中出现的许多现象不能用确定性动力学（在没有噪声的情况下）来表征。实际上，在经典系统中引入噪声可以极大地改变它们的动力学行为。尽管随机系统由于其许多应用而非常受欢迎，但到目前为止，即使对于低维系统，对其定性行为的理论一般理解也有限。这个博士学位的主要重点将是研究吸收随机系统，即以不被某些罕见事件杀死为条件的系统。例如，这可以模拟股票低于一定价格（障碍期权）的演变，材料破裂前裂缝的扩展，或者种群灭绝前的增长。由于这些不同的应用，吸收系统最近吸引了一个非常活跃的社区的兴趣。目的和目的：本研究的范围有三个方面：1。通过研究条件李雅普诺夫指数等工具的存在性和相关性，为研究吸收系统的动态行为建立理论框架。这些对象用来描述这些系统的某些特性，如可预测性或混沌性。2 .研究一些以生存为条件的自然系统的长期、现在和过去的统计数据，这些系统的结果是未知的。设计和实现计算方法来模拟这样的系统，以演示上述工具的实用性（以及相关性）。研究方法的新颖性：本项目建立在条件随机动力系统理论的最新进展基础上，由导师和合作者共同完成。该项目将利用在随机分析、动力系统和遍历理论方面发展起来的最先进技术。这个博士的研究是由顶尖数学机构对吸收系统日益增长的兴趣所激发的。该项目由Martin Rasmussen博士和Jeroen Lamb教授担任导师，Matheus De Castro、Dennis Chemnitz和Maximilian Engel博士担任合作者。这项研究也得到了EPSRC随机系统数学博士培训中心的支持：分析、建模和算法。该项目属于EPSRC数学科学主题和非线性系统，数值分析，统计和应用概率研究领域。

This research aims at expanding our understanding of random dynamical systems, i.e., systems whose underlying evolution laws are driven by noise. A thorough understanding of such systems is crucial as many phenomena arising in finance, ecology or climate science cannot be characterised by deterministic dynamics (in the absence of noise). Indeed, the introduction of noise to classical systems can drastically change their dynamical behaviour. Although random systems are very popular due to their many applications, there is so far, a limited theoretical general understanding of their qualitative behaviour even for low-dimensional systems.The main focus of this PhD will be on studying absorbed random systems, that is systems conditioned on not being killed by some rare event. This could for instance model, the evolution of a stock below a certain price (barrier option), the expansion of a crack before a material breaks down or the growth of a population before its extinction. Due to these varied applications, absorbed systems have recently attracted the interest of a very active community.Aims and objectives:The scope of this research is threefold:1. Develop a theoretical framework for the study of the dynamic behaviour of absorbed systems by investigating the existence and relevance of tools such as conditioned Lyapunov exponents. Such objects serve to characterise some qualities of these systems such as predictability or chaos.2. Study the long-term present and past statistics of systems conditioned on survival for some natural systems for which such results are not known.3. Design and implement computational methods to simulate such systems to the end of demonstrating the practicality (and thus relevance) of the aforementioned tools.Novelty of the research methodology:This project builds on recent advances in the theory of conditioned random dynamical systems achieved with the supervisors and collaborators. The project will make use of state-of-the-art techniques developed in stochastic analysis, dynamical systems and ergodic theory. The research of this PhD is stirred by a growing interest in absorbed systems at top mathematical institutions.The project has Dr Martin Rasmussen and Prof. Jeroen Lamb as supervisors and Matheus De Castro, Dennis Chemnitz and Dr Maximilian Engel as collaborators.This research is also supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Algorithms.This project falls within the EPSRC Mathematical Sciences theme and the non-linear systems, numerical analysis, statistics and applied probability research areas.