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 non-zero 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 that for the physical model. Such sequences of controls and solutions are not usually tight in the Skorokhod topology, the usual one in weak-covergence work. 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, and to show that there is an optimal control. For many applications, such as the convergence of numerical approximations to the limit model, one needs to be able to characterize and approximate optimal controls by discrete-valued classical singular controls, a problem made more difficult when there are MSIs. This paper deals with such problems, when the driving noise is either a Wiener or a wide-band-width noise process. The treatment of the first case uses an adaptation of the versatile time-stretching method, while the treatment of the wide-band-width noise case uses that together with the powerful perturbed test function method.