Finite dimensional Realizations of Stochastic Equations
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, Bj\"ork 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 Bj\"ork-Svensson setting. A substantial part of this paper is devoted to analysis on Fr\'echet 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.