The class of arbitrage free interest-rate models introduced by Heath, Jarrow and Morton in the 1980s has become the standard mathematical framework for interest rate theory since. In the recent years many researchers have approached HJM models from the viewpoint of infinite dimensional stochastic analysis, viewing them as evolution equations in the space of forward rate curves.
This monograph, which is based on the PhD thesis of its author at ETH Zurich, contains a rigorous mathematical presentation
of this point of view on HJM models plus a detailed discussion of the so-called "consistency problem": when is a given parametric family of forward rate curves invariant under an evolution specified by an HJM model?Chapter 2 "Stochastic equations in infinite dimensions" is interesting in itself as a minimal and economical introduction to this topic without needlessly complicated formalism.
For example the infinite dimensional Wiener process is introduced as a sequence of independent scalar Wiener processes
and everything is understandable for a student who knows one dimensional Brownian motion.
Chapters 4 and 5 define a possible mathematical framework for viewing HJM models as evolution equations. This discussion, some of which is original work of the author, is what is missing (or even wrong) in many other works on this topic. I found this chapter clear and helpful; it is the best part of the book.
Chapters 6 and 7 discuss recent researh by Bjork, Christensen, Zabczyk and the author on invariant finite dimensional manifolds for HJM models. Although this is the main focus of the thesis it seems less interesting to me since I never saw any real
motivation of this problem from financial applications.
Some weak points of the book are: total absence of any empirical data, no figures (there is not even a single figure of what a forward curve looks like), no numerical examples showing how "bad" inconsistency can be, ...Also no application is given...But then these are problem typical of the whole corpus of (continental) European literature on mathematical finance.
Another point is the relation between the "infinite dimensional" viewpoint and the "random field" interest rate models. This is discussed in some parts of the text (see Chap 5) but only briefly.
