随机方程的有限维实现
Finite dimensional Realizations of Stochastic Equations
摘要：
本文讨论Heath-Jarrow-Morton利率模型的有限维（马尔可夫）实现。我们考虑一个d维驱动布朗运动和随机波动率结构，这是当前远期利率的非退化光滑泛函。在最近的一篇论文中，Björk和Svensson给出了在特定的希尔伯特空间中FDR存在的充分和必要条件。我们扩展了他们的框架，提供了新的结果的几何隐含FDR和分类。特别是，我们证明了他们的猜想，每个短率实现是2维的。更一般地，我们表明，所有通用的FDR至少（d+1）维，所有通用的FDR是仿射的。作为一个例子，我们画了一个利率模型，这是很好的斯文森曲线拟合方法。这些结果不能在比约克-斯文森设置中获得。本文的一个重要部分是致力于分析的Fréchet空间，在那里我们得到一个Frobenius定理。虽然我们只考虑随机方程的HJM框架，许多结果结转到一个更一般的设置。
英文摘要：
This paper discusses finite-dimensional (Markovian) realizations (FDRs) for Heath-Jarrow-Morton interest rate models. We consider a d-dimensional driving Brownian motion and stochastic volatility structures that are non-degenerate smooth functionals of the current forward rate. In a recent paper, Björk and Svensson give sufficient and necessary conditions for the existence of FDRs within a particular Hilbert space setup. We extend their framework, provide new results on the geometry of the implied FDRs and classify all of them. In particular, we prove their conjecture that every short rate realization is 2-dimensional. More generally, we show that all generic FDRs are at least (d+1)-dimensional and that all generic FDRs are affine. As an illustration we sketch an interest rate model, which goes well with the Svensson curve-fitting method. These results cannot be obtained in the Björk-Svensson setting. A substantial part of this paper is devoted to analysis on Fréchet spaces, where we derive a Frobenius theorem. Though we only consider stochastic equations in the HJM-framework, many of the results carry over to a more general setup.
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