Controlling inflation: The infinite horizon case

This paper studies the two-dimensional singular stochastic control problem over an infinite time-interval arising when the Central Bank tries to contain the inflation by acting on the nominal interest rate. It is shown that this problem admits a variational formulation which can be differentiated (in some sense) to lead to a stochastic differential game with stopping times between the conservative and the expansionist tendencies of the Bank. Substantial regularity of the free boundary associated to the differential game is obtained. Existence of an optimal policy is established when the regularity of the free boundary is strengthened slightly, and it is shown that the optimal process is a diffusion reflected at the boundary.