Path-to-signature isometries with applications to modelling the long-term dynamics of complex systems

路径到特征等距及其在复杂系统长期动态建模中的应用

• 批准号: EP/W00707X/1
• 负责人: Anastasia Papavasiliou
• 金额: $ 6.53万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW00707X%2F1
• 关键词: Path; signature; isometries; applications; modelling

Almost all natural and man-made processes behave differently at different time scales. For example, if we plot the temperature in Coventry on a minute-by-minute scale over an hour we would expect to see small smooth changes with no clear trend. On the other hand, we would expect the weekly temperature over a year to exhibit large fluctuations and a seasonal trend. To capture the behaviour of temperature changes on all scales, we would need to use complex, high-dimensional dynamical systems. However, such systems can be extremely inefficient, which is why coarse-grained models, describing the long-term dynamics, are often used instead. Our aim is to address the problem of fitting coarse-grained models to data. The main challenge is that coarse-grained models, while successful in providing a good approximation of the long-term dynamics, often fail to capture the fine-scale properties of the system. Typically, coarse-grained models exhibit a rougher behaviour (e.g. equivalent to Brownian motion) than the full complex systems, which in the very fine-scale are usually of bounded variation. As a result, direct use of standard estimators can lead to wrong results, unless the mismatch between model and data is carefully addressed. The main limitation of current methodology is that it depends on explicit knowledge of the scale separation parameter, which allows us to use data in a scale compatible with the coarse-grained model. However, this information is usually not available.We will construct a new estimator based on a rapidly developing tool known as the rough path signature, which is a purpose-built tool for stochastic models with multiscale behaviour. The rough path signature is a sequence where the first term describes the behaviour of the model at a smooth scale, while the second term sees the finer Brownian scale, and so on. The limiting asymptotics of the signature capture the behaviour of the model at all scales, and it is possible to extract the behaviour in a single scale by appropriate normalisation. Our goal will be to identify the normalization that will lead to the extraction of the Brownian scale, thus providing an estimator for the diffusion coefficient, by making implicit use of the scale separation exhibited by the data. The key theoretical underpinning of this estimator is a recently discovered formula by the co-I and his collaborator for extracting the behaviour of a path at the smooth and Brownian scales from the signature. The second objective of the project is to extend these results to the bounded variation scale. A fundamental difficulty has been how to move beyond the assumption of continuous derivative. However, the co-I and his collaborator have recently managed to achieve this in a class of two-dimensional models. We will build on this discovery to show a general formula for extracting the bounded variation behaviour from the signature in the second objective. The proposed research is a first step towards a much larger research programme. One of the main advantages of our approach is that signature-based estimators should naturally generalise to all scales and, consequently, more general models. In order to fully develop the signature as a standard tool in multiscale modelling, we must extend this "scale-extraction" result to all scales. This will require a systematic methodology for the identification of the appropriate normalisation constant, both in the context of exact models and coarse-grained models.

几乎所有自然和人造过程在不同的时间尺度上表现都不同。例如，如果我们在一个小时内以分钟为单位绘制考文垂的温度，我们预计会看到小的平滑变化，没有明显的趋势。另一方面，我们预计一年中的每周气温会表现出较大的波动和季节性趋势。为了捕捉所有尺度上的温度变化行为，我们需要使用复杂的高维动力系统。然而，这样的系统可能效率极低，这就是为什么经常使用描述长期动态的粗粒度模型。我们的目标是解决粗粒度模型与数据的拟合问题。主要挑战是粗粒度模型虽然成功地提供了长期动态的良好近似，但通常无法捕获系统的精细尺度属性。通常，粗粒度模型表现出比完整复杂系统更粗糙的行为（例如，相当于布朗运动），而完整复杂系统在非常精细的尺度上通常具有有限的变化。因此，除非仔细解决模型和数据之间的不匹配问题，否则直接使用标准估计器可能会导致错误的结果。当前方法的主要限制是它依赖于尺度分离参数的显式知识，这使得我们能够使用与粗粒度模型兼容的尺度中的数据。然而，这些信息通常不可用。我们将基于快速开发的工具（称为粗糙路径签名）构建一个新的估计器，该工具是针对具有多尺度行为的随机模型的专用工具。粗糙路径签名是一个序列，其中第一项描述了模型在平滑尺度上的行为，而第二项则描述了更精细的布朗尺度，依此类推。签名的极限渐近捕获了模型在所有尺度上的行为，并且可以通过适当的归一化来提取单个尺度的行为。我们的目标是确定导致布朗尺度提取的归一化，从而通过隐式使用数据所显示的尺度分离来提供扩散系数的估计器。 该估计器的关键理论基础是 co-I 和他的合作者最近发现的一个公式，用于从签名中提取平滑和布朗尺度下的路径行为。该项目的第二个目标是将这些结果扩展到有界变化范围。一个根本的困难是如何超越连续导数的假设。然而，我的同事和他的合作者最近成功在一类二维模型中实现了这一目标。我们将在此发现的基础上展示从第二个目标中的签名中提取有界变化行为的通用公式。拟议的研究是迈向更大规模研究计划的第一步。我们的方法的主要优点之一是基于签名的估计器应该自然地推广到所有尺度，从而推广到更通用的模型。为了充分发展签名作为多尺度建模的标准工具，我们必须将这种“尺度提取”结果扩展到所有尺度。这将需要一种系统的方法来识别适当的归一化常数，无论是在精确模型还是粗粒度模型的背景下。

项目成果

On the Lack of Gaussian Tail for Rough Line Integrals along Fractional Brownian Paths

关于沿分数布朗路径的粗线积分缺乏高斯尾部的问题

• 发表时间: 2022
• 作者: H Boedihardjo

Estimating the volatility of highly traded stocks from the signature

从签名估计交易量大的股票的波动性

• 发表时间: 2022
• 作者: Louis March