Optimal Inspections in a Stochastic Control Problem with Costly Observations, II

A Brownian motion ξ(t) is developing in time with cost f(ξ(t)) per unit time. It is assumed that ξ(t) = (x(t), y(t)) where x(t) and y(t) are independent Brownian motions. The component x(t) is being continuously observed, whereas the position of y(t) can be discovered only by making observations at random times σ_n with incurred cost β(ξ(σ_n)). Thus, σ_n is a stopping time with respect to the σ-field generated by x(t), t ≥ 0 and the random variables y(σ₁), …, y(σ_n−1). A set A is given, and at the time σ_n the following policy is executed: (a) continue with the process until the next inspection, if y(σ_n) ∈ A, (b) stop and shut off the process with cost γ(ξ(σ_n) if y(σ_n) ∉ A. The problem considered in this paper is that of finding an optimal sequence of inspections {σ_n} This is done by first transforming the stochastic problem into a free boundary problem in analysis and then studying the latter.