On systems and control concepts in linear interest rate theory

This contribution discusses the concept of stochastic controllability within the framework of linear interest rate models of Heath-Jarrow-Morton-Musiela (HJMM) type that may be represented by an infinite dimensional stochastic differential equation. Despite the fact that not all such models are controllable, we nonetheless investigate the possibility of influencing the drift term of the aforementioned differential equation by a particular choice of control function. As a consequence, the primary purpose of our study is to determine necessary and sufficient conditions for the stochastic controllability of a special subclass of the aforementioned models. In particular, we find 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. In order to address this problem we introduce deterministic and stochastic controllability operators related to such interest rate models and solve a linear regulator problem associated with the minimum energy principle.