arXiv subjects

Damir Filipovic

Publications and source records attributed to Damir Filipovic.

21 records · Page 2Linked to original sources

arXiv2004

Regularity of finite-dimensional realizations for Evolution Equations

We show that a continuous local semiflow of $C^k$-maps on a finite-dimensional $C^k$-manifold M can be embedded into a local $C^k$-flow on M under some weak (necessary) assumptions. This result is applied to an open problem in [fil/tei:01]. We prove that finite-dimensional realizations for interest rate models are highly regular objects, namely given by submanifolds M of $D(A^{\infty})$, where A is the generator of a strongly continuous semigroup.

math.FA↗

arXiv2002

On Finite-dimensional Term Structure models

In this paper we provide the characterization of all finite-dimensional Heath--Jarrow--Morton models that admit arbitrary initial yield curves. It is well known that affine term structure models with time-dependent coefficients (such as the Hull--White extension of the Vasicek short rate model) perfectly fit any initial term structure. We find that such affine models are in fact the only finite-factor term structure models with this property. We also show that there is usually an invariant singular set of initial yield curves where the affine term structure model becomes time-homogeneous. We also argue that other than functional dependent volatility structures -- such as local state dependent volatility structures -- cannot lead to finite-dimensional realizations. Finally, our geometric point of view is illustrated by several examples.

math.PR↗

arXiv2001

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, 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.

math.PR↗