A note on the long rate in factor models of the term structure

In this paper, we consider factor models of the term structure based on a Brownian filtration. We show that the existence of a nondeterministic long rate in a factor model of the term structure implies, as a consequence of the
Dybvig–Ingersoll–Ross theorem, that the model has an equivalent representation in which one of the state variables is nondecreasing. For two-dimensional factor models, we prove moreover that if the long rate is nondeterministic, the yield curve flattens out, and the factor process is asymptotically nondeterministic, then the term structure is unbounded. Finally, we provide an explicit example of a three-dimensional affine factor model with a nondeterministic yet finite long rate in which the volatility of the factor process does not vanish over time.