Stochastic Controllability of Linear Interest Rate Models

Abstract

We consider the controllability problem for the linear HeathJarrow-Morton-Musiela (HJMM) interest rate model that is realized by an infinite-dimensional stochastic differential equation (SDE). Although it is clear that interest rates are not generally controllable, the objective of our paper is nevertheless to establish necessary and sufficient conditions for the stochastic controllability of a special subclass of the aforementioned models. In this process we determine a control that transfers the said model from an arbitrary interest rate to any other interest rate in the state space of forward rate curves. Our method of solving this problem involves a consideration of the deterministic and stochastic controllability operators related to the aforementioned SDE and their resolvents and a regulator problem associated with the minimum energy principle. In this regard, a formula for a minimizing control is given explicitly in terms of an invertible deterministic controllability operator. Also, we briefly comment on connections between the main results of the paper and the related Ho-Lee, Hull-White and Cox-Ingersoll-Ross interest rate models.