On Finite Dimensional Realizations of Forward Price Term Structure Models

In this paper we study a fairly general Wiener driven model for the term structure of forward prices. The model, under a fixed martingale measure, Q, consists of two infinite dimensional stochastic differential equations (SDEs). The first system is a standard HJM model for (forward) interest rates, driven by a multidimensional Wiener process W. The second system is an infinite SDE for the term structure of forward prices on some specified underlying asset driven by the same W. We are primarily interested in the forward prices. However, since for any fixed maturity, T, the forward price process is a martingale under the T-forward neutral measure, the zero coupon bond volatilities will enter into the drift part of the SDE for these forward prices. The interest rate system is, thus, needed as input into the forward price system. Given this setup we use the Lie algebra methodology of Björk et al. to investigate under what conditions on the volatility structure of the forward prices and/or interest rates, the inherently (doubly) infinite dimensional SDE for forward prices can be realized by a finite dimensional Markovian state space model.