Stationary stochastic control for Itô processes

Consider a real-valued Itô process X(t) = x + .0t.(s)ds + .0t.(s)dW(s) + A(t) driven by a Brownian motion {W(t) : t > 0}. The controller chooses the real-valued progressively measurable processes ., . and A subject to constraints |.(t)| . .0(X(t-)) and |.(t)| . .0(X(t-)), where the functions .0 and .0 are given. The process A is a bounded variation process and |A|(t) represents its total variation on [0,t]. The objective is to minimize the long-term average cost lim supT..(1/T)E[|A|(T) + .0Th(X(s))ds], where h is a given nonnegative continuous function. An optimal process X* is determined. It turned out that X* is a reflecting diffusion process whose state space is a finite interval [a*, b*]. The optimal drift and diffusion controls are explicitly derived and the optimal bounded variation process A* is determined in terms of local-time processes of X* at the points a* and b*.