Structure-Preserving Integrators for Lévy-Driven Stochastic Systems

Levy 驱动随机系统的结构保持积分器

• 批准号: EP/Y033248/1
• 负责人: A Wiese
• 金额: $ 9.04万
• 依托单位: Heriot-Watt University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY033248%2F1
• 关键词: Structure; Preserving; Integrators; vy; Driven

The fundamental aim of this proposal is to further the understanding of stochastic differential equations driven by Lévy processes and their algebraic structures, the design and analysis of novel structure-preserving integrators when the system is constrained to evolve on a manifold and the modelling of such systems. In many applications the evolution of quantities is random in nature. Key stochastic processes for describing the random driving force are Wiener processes as models for Gaussian random noise, and more generally Lévy processes, as generalizations of Wiener processes in applications when randomness cannot always be captured accurately by Gaussian random factors. Imagine zooming into a time series of financial data by increasing the frequency of observations. In econometric studies it has been observed that on the smaller time scale data will typically exhibit larger fluctuations and hence non-Gaussian behaviour. With the increasing amount of data we are now able to observe and to process, more complex stochastic differential equations driven by Lévy processes are becoming increasingly more important. Applications are numerous, including in climate science, where changes in some weather patterns have been observed to occur in jumps, in the modelling of financial quantities such as stock prices and interest rates, or in biology for example in models for the movement of cells. It is typical that stochastic differential equations describing these evolutions, even in the continuous case, have no known solution as a given function of the driving stochastic processes. The design and analysis of numerical integrators is thus pivotal in modelling and in understanding and analysing these equations. This challenge is compounded when the solution to the stochastic differential system is known to evolve on a manifold, and even further when dealing with the jump discontinuities of a Lévy process. Standard Taylor series-based numerical integrators are not designed to generate approximate solutions that remain on the manifold, and projecting these approximate solutions onto the manifold may not be feasible or efficient. The proposed research programme aims to address this challenge. It is based on recent findings of the proposer and collaborators that link in an intrinsic way stochastic differential equations and their integrators with algebraic structures that encompass key properties of the stochastic system under consideration and that enable the design of novel efficient integrators that are more accurate than standard Taylor series expansion schemes in Gaussian- and Lévy-driven stochastic differential equations and the design of structure-preserving methods for continuous stochastic differential equations.The current research project aims to extend these methods to stochastic differential equations driven by Lévy processes. It will bring together ideas from stochastic analysis, stochastic differential geometry, algebra, and numerical analysis. The project will further the understanding of stochastic differential equations driven by Lévy processes and evolving on manifolds and the intrinsic link to their algebraic structure, and it will develop novel generic structure-preserving numerical methods for solving Lévy-driven models that were previously not solvable.

这个建议的基本目的是进一步了解随机微分方程驱动的Lévy过程和他们的代数结构，设计和分析新颖的结构保持积分器时，系统被约束到发展的流形和建模这样的系统。在许多应用中，量的演化本质上是随机的。描述随机驱动力的关键随机过程是作为高斯随机噪声模型的维纳过程，更一般的是作为维纳过程在随机性不能总是被高斯随机因子准确捕获的应用中的推广的Lévy过程。想象一下，通过增加观察的频率来放大金融数据的时间序列。在计量经济学研究中，已经观察到，在较小的时间尺度上，数据通常会表现出较大的波动，因此具有非高斯行为。随着我们现在能够观察和处理的数据量的增加，由Lévy过程驱动的更复杂的随机微分方程变得越来越重要。应用范围很广，包括在气候科学中观察到某些天气模式的变化以跳跃的方式发生，在股票价格和利率等金融量的建模中，或在生物学中，例如在细胞运动模型中。这是典型的随机微分方程描述这些演变，即使在连续的情况下，没有已知的解决方案，作为一个给定的函数的驱动随机过程。因此，数值积分器的设计和分析是建模和理解和分析这些方程的关键。当已知随机微分系统的解在流形上演化时，甚至在处理Lévy过程的跳跃不连续性时，这一挑战变得更加复杂。标准的基于泰勒级数的数值积分器不是设计来生成保留在流形上的近似解的，并且将这些近似解投影到流形上可能是不可行或不有效的。拟议的研究方案旨在应对这一挑战。它基于提议者和合作者的最新发现，这些发现以内在的方式将随机微分方程及其积分器与代数结构联系起来，这些代数结构包含所考虑的随机系统的关键属性，并且使得能够设计比高斯和Lévy驱动的随机微分方程中的标准泰勒级数展开方案更准确的新型有效积分器，以及结构的设计。连续型随机微分方程的保持方法，本研究计划旨在将这些方法推广到由Lévy过程驱动的随机微分方程。它将汇集来自随机分析，随机微分几何，代数和数值分析的想法。该项目将进一步了解由Lévy过程驱动的随机微分方程，并在流形上演化及其代数结构的内在联系，并将开发新的通用结构保持数值方法来解决Lévy驱动的模型，这些模型以前无法解决。