Structure-Preserving Integrators for Lévy-Driven Stochastic Systems

这一建议的基本目的是加深对L过程驱动的随机微分方程及其代数结构的理解，当系统被约束在流形上演化时，新的保结构积分器的设计和分析，以及这类系统的建模。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维纳过程，作为高斯随机噪声的模型，更广泛地说，是L过程，作为维纳过程在应用中的推广，当随机性不总是被高斯随机因子准确捕获时。想象一下，通过增加观察频率来放大金融数据的时间序列。在计量经济学研究中，人们观察到，在较小的时间尺度上，数据通常会表现出较大的波动，从而表现出非高斯行为。随着我们现在能够观察和处理的数据量越来越大，由L过程驱动的更复杂的随机微分方程变得越来越重要。应用很多，包括在气候科学中，观察到一些天气模式的变化是跳跃发生的，在股票价格和利率等金融数量的建模中，或者在生物学中，例如在细胞运动的模型中。通常，描述这些演化的随机微分方程，即使在连续的情况下，也不存在作为驱动随机过程的给定函数的已知解。因此，数值积分器的设计和分析对于建模以及理解和分析这些方程至关重要。当随机微分系统的解在流形上演化时，甚至当处理Lévy过程的跳跃不连续性时，这一挑战变得更加复杂。基于泰勒级数的标准数值积分器不是被设计成生成保留在流形上的近似解，并且将这些近似解投影到流形上可能是不可行或不高效的。拟议的研究方案旨在应对这一挑战。它基于作者和合作者的最新发现，将随机微分方程及其积分器与包含所讨论随机系统的关键性质的代数结构内在地联系在一起，从而能够设计出比Gauss型和Lé型随机微分方程中的标准泰勒级数展开式更精确的新的有效积分器，并设计出连续随机微分方程的结构保持方法。它将汇集随机分析、随机微分几何、代数和数值分析的思想。该项目将进一步加深对由Lévy过程驱动的在流形上发展的随机微分方程及其与其代数结构的内在联系的理解，并将开发新的一般结构保持的数值方法来求解以前不可解的Lévy驱动的模型。

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.