Stationary stochastic control for Itô processes

Abstract

Consider a real-valued Itô process X(t) = x + ∫0tμ(s)ds + ∫0tσ(s)dW(s) + A(t) driven by a Brownian motion {W(t) : t > 0}. The controller chooses the real-valued progressively measurable processes μ, σ and A subject to constraints |μ(t)| ≤ μ0(X(t-)) and |σ(t)| ≥ σ0(X(t-)), where the functions μ0 and σ0 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*.