Long-term average control of a continuous,monotone process

We analyze optimal control problems for systems subject to random deterioration and failure. The system is replaced at failure and our objective is to optimize the utilization of the system between failures. The problems are new in that the payoff depends on the running maximum of a diffusion. This provides an intuitively appealing model for naturally monotone phenomena such as wear. The long-term average control problem is reduced to a family of simpler, single-cycle problems, a formula for the invariant measure for the (controlled) process is determined and a computational scheme based on the decomposition and formula is given.