Minimal realizations of interest rate models

We consider interest rate models where the forward rates are allowed to be driven by a multidimensional Wiener process as well as by a marked point process. Assuming a deterministic volatility structure, and using ideas from systems and control theory, we investigate when the inputoutput map generated by such a model can be realized by a finite dimensional stochastic differential equation. We give necessary and sufficient conditions, in terms of the given volatility structure, for the existence of a finite dimensional realization and we provide a formula for the determination of the dimension of a minimal realization. The abstract state space for a minimal realization is shown to have an immediate economic interpretation in terms of a minimal set of benchmark forward rates, and ∗Support from ITM and CNR, as well as the hospitality and support of the Laboratory of Actuarial Mathematics, Copenhagen University, is gratefully acknowledged. The authors wish to thank an anonymous referee for a number of helpful comments.