Consistency problems for Heath-Jarrow-Morton : interest rate models / Damir Filipovic

This topical research monograph focuses on intrinsic properties of finite-dimensional manifolds of (forward) curves, which can be generated by factor models of the term structure of interest rates within the Heath-Jarrow-Morton framework, and their relation to empirical curve-fitting methods, which are being used to obtain a current forward curve. After a well-written introduction, the technical toolbox for this discussion, stochastic analysis in infinite dimensions, is outlined in Chapter 2. The following chapter provides the rigorous setup for factor models described within the framework of parametrized families of smooth curves (which are typically used for fitting initial forward curves). As a first main result a general characterisation of diffusion models which, together with particular smooth curves, provide arbitrage-free interest rate models is given. In Chapter 4 the Heath-Jarrow-Morton methodology is extended to incorporate an infinite-dimensional driving Brownian motion. The change to the Musiela parametrization is performed and the Heath-Jarrow-Morton description of the forward rate as a system of infinitely many Itô processes is represented as one dynamical system in infinite dimensions within a Hilbert space framework. This key stochastic equation is further analysed in Chapter 5, where it is also shown that classical models fit well into the framework. Chapter 6 provides further technical details on finite-dimensional submanifolds, which are used in Chapter 7 to provide consistency results for the common families of curves.
To conclude, I found the book a very useful addition to the current literature. It brings together new and important results relating the problem of empirical curve-fitting methods with underlying modelling assumptions within widely used term-structure models. (MathSciNet)