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Minimal realizations in interest rate models. (English) Zbl 0947.60051
The purpose of this paper is to investigate when and how a given, a priori infinite-dimensional, forward rate Heath-Jarrow-Morton type model with deterministic volatility can be generated by a finite-dimensional model. The authors deal with the Musiela parameterization and study only a linear case. Using ideas and methods from system and control theory, they succeed to solve the mentioned problems by studying the transfer function of an associated deterministic system. Thus, necessary and sufficient conditions for the existence of a finite-dimensional realization are obtained in terms of the given volatility structure and a formula for the dimension of a minimal realization is established. It is shown that the abstract state space for a minimal realization has an immediate economic interpretation in terms of a minimal set of benchmark forward rates. As a simple illustration of this theory the explicit formulas for bonds and prices are obtained. Finally, the results proved for systems driven by a multidimensional Wiener process are extended for models driven by a marked point process. In general, the authors consider their results as a first step towards an application of stochastic realization theory to the area of interest rates.

MSC:
60G44 Martingales with continuous parameter
91B70 Stochastic models in economics
93B15 Realizations from input-output data
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