Abstract:
We consider a parametrization of the Heath-Jarrow-Morton (HJM) family of term
structure of interest rate models that allows a finite-dimensional Markovian representation
of the stochastic dynamics. This parametrization results from letting the volatility
function depend on time to maturity and on two factors: the instantaneous spot rate
and one fixed-maturity forward rate. Our main purpose is an estimation methodology for
which we have to model the observations under the historical probability measure. This
leads us to consider as an additional third factor the market price of interest rate risk, that
connects the historical and the HJM martingale measures. Assuming that the information
comes from noisy observations of the fixed-maturity forward rate, the purpose is to
estimate recursively, on the basis of this information, the three Markovian factors as well as the parameters in the model, in particular those in the volatility function. This leads
to a nonlinear filtering problem, for the solution of which we describe an approximation
methodology, based on time discretization and quantization. We prove the convergence
of the approximate filters for each of the observed trajectories.
