Shapes of the Term Structure of Interest Rates

利率的期限结构——以收益率曲线或远期曲线的形式概括——是最基本的经济指标之一。它的形状编码了短期和长期投资偏好、对流动性的渴望以及对央行决策和总体经济前景的预期等重要信息。因此，任何期限结构的数学模型都会自然而然地提出这样一个问题：该模型能够再现收益率曲线和远期曲线的哪种形状，以及它们出现的频率是多少？虽然这个问题在一维仿射期限结构模型中已经得到了较早和结论性的回答，但在更相关的二维情况下，系统的结果仅在最近几年才出现。在最近的一项突破中，引入了包络的几何方法来全面分析和分类二维Vasicek模型中所有可能的期限结构形状。在本项目中，我们将利用包络的方法解决大量相关利率模型中不同形状的形状分类、状态空间分割和长期频率计算问题，远远超出Vasicek模型。虽然Vasicek模型由于其高斯性质和Riccati ode的线性化而相对简单，但在将包络方法扩展和适应其他具有或不具有跳跃的仿射期限结构模型时，还需要克服大量的挑战。当考虑非仿射或时间非齐次仿射模型时，会出现额外的挑战。而不是包络的线族，一族沉浸在模型状态空间必须考虑。在对奇异点和自交点进行分类时，将使用总正性包络理论（由塞缪尔·卡林在20世纪60年代首创）。

The term structure of interest rates -- summarized in the form of the yield curve or the forward curve -- is one of the most fundamental economic indicators. Its shape encodes important information on the preferences for short- vs. long-term investments, the desire for liquidity and on expectations of central bank decisions and the general economic outlook. It is therefore a natural question -- to be asked of any mathematical model of the term structure -- which shapes of yield curves and forward curves the model is able to reproduce and with which frequency they appear. While this question has been answered earlier and conclusively for one-dimensional affine term structure models, systematic results on the more relevant two-dimensional case have appeared only in the last few years. In a recent break-through, the geometric approach of envelopes has been introduced to completely analyze and classify all possible term structure shapes in the two-dimensional Vasicek model. In this project, we will leverage the method of envelopes to solve the problems of shape classification, state-space segmentation and calculation of long-run frequencies of different shapes in a large number of relevant interest rate models, well beyond the Vasicek model. While the Vasicek model is relatively simple due to its Gaussian nature and linearization of Riccati ODEs, substantial challenges will have to be overcome when extending and adapting the envelope approach to other affine term structure models with and without jumps. Additional challenges appear when non-affine or time-inhomogeneous affine models are considered. Instead of envelopes of famillies of lines, families of one-manifold immersed in the models state space have to be considered. When classifying singular points and self-intersections of the envelope theory of total positivity (pioneered by Samuel Karlin in the 1960ies) will be used.