(Joint work with Lars Svensson, Department of Mathematics, Royal Institute of Technology, Stockholm.)

Abstract:

We consider interest rate models of
Heath-Jarrow-Morton type, where the forward rates are
driven by a multi-dimensional Wiener process, and where the
volatility is allowed to be an arbitrary smooth functional
of the present forward rate curve. Using ideas from
differential geometry as well as from systems and control
theory, we investigate when the forward rate process can be
realized by a finite dimensional Markovian state space
model, and we give general necessary and sufficient
conditions, in terms of the volatility structure, for the
existence of a finite dimensional realization. A number of
concrete applications are given, and most previously known
realization results for time homogenous Wiener driven
models are included and extended. As a special case we give
a general and easily applicable necessary and sufficient
condition for when the induced short rate is a Markov
process. In particular we show that the only forward rate
models, with short rate dependent volatility structures,
which generically give rise to a Markovian short rate are
the affine ones. These models are thus the only generic
short rate models from a forward rate point of view.