MODELING AND APPROXIMATIONS FOR STOCHASTIC SYSTEMS WITH STATE-DEPENDENT SINGULAR CONTROLS AND WIDE-BAND NOISE

Singular controls are usually idealizations of controls that are large and occur over a small but nonzero amount of time, or else are limits of a sequence of small impulses occurring close together. When these control terms are multiplied by a function of the state, it is what we call "state-dependence." Because of this state-dependence, there usually are problems in using standard weak convergence arguments to validate the limits and show that the costs for the limit well approximate those for the physical model. Such sequences of controls and solutions are not usually tight in the Skorokhod topology, the usual one used in weak convergence work. We are concerned with such problems when the driving noise is a wide bandwidth process, which greatly complicates the analysis. Owing to the state-dependence, in the limit we can get what we call multiple simultaneous impulses (MSIs), each of which is characterized as the limit of a sequence of state-dependent control actions over a time interval that collapses to zero, and great care is needed to characterize the limit controls and process paths to show that there is an optimal control for the limit model and to determine which is nearly optimal for the prelimit physical models. The treatment uses an adaptation of the versatile time-stretching method to deal with the controls, together with the powerful perturbed test function method, to handle the wide bandwidth noise.