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

A Brownian motion z ( t ) is known to be developing in time with cost f ( z ( t )) per unit lime. One chooses a sequence of times (sigma) n in which to inspect the motion. If it is discovered that z ((sigma) n ) is in a given set A then we continue with the motion, whereas if z ((sigma) n ) is not in A then we shut off the motion with cost y ( z ((sigma) n )). Each observation at t = (sigma) n costs (beta)( z ((sigma) n )). In this paper we find a sequence of inspections for which the total cost is minimum. This is done, in Part I, by reducing the stochastic problem lo a free boundary problem in analysis (called a quasi variational inequality), which is then solved. In Part II we solve a similar problem whereby, instead of shutting off the motion when z ((sigma) n ) is not in A , we innovate it by bringing it to the origin at some cost y ( z ((sigma) n )). Finally, in Part III we consider the case where z ( t ) can only be observed (at the times (sigma) n ) with an error.