Stationary stochastic control for Ito processes

Consider a real-valued Ito process X(t) = x + integral(0)(t) mu(s) ds+f(0)(t) sigma(s) dW(s) + A(t) driven by a Brownian motion {W(t) : t > 0}. The controller chooses the real-valued progressively measurable processes mu, sigma and A subject to constraints |mu(t)| less than or equal to mu(0)(X(t-)) and |sigma(t)| greater than or equal to sigma(0)(X(t-)), where the functions mu(0) and sigma(0) are given, The process A is abounded variation process and |A|(t) represents its total variation on [0, t]. The objective is T to minimize the long-term average cost lim sup(T-->infinity)(1/T)E[|A|(T) + integral(0)(T) h(X(s))ds], where his 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*.