Degenerate Variance Control in the One-dimensional Stationary Case

Abstract

We study the problem of stationary control by adaptive choice of the diffusion coefficient in the case that control degeneracy is allowed and the drift admits a unique, asymptotically stable equilibrium point. We characterize the optimal value and obtain it as an Abelian limit of optimal discounted values and as a limiting average of finite horizon optimal values, and we also characterize the optimal stationary strategy. In the case of linear drift, the optimal stationary value is expressed in terms of the solution of an optimal stopping problem. We generalize the above results to allow unbounded cost functions.