Jump-Diffusions with Controlled Jumps: Existence and Numerical Methods

A comprehensive development of effective numerical methods for stochastic control problems in continuous time, for reflected jump-diffusion models, is given in earlier work by the author. While these methods cover the bulk of models which have been of interest to date, they do not explicitly deal with the case where the jump itself is controlled in the sense that the value of the control just before the jump affects the distribution of the jump. We do not deal explicitly with the numerical algorithms but develop some of the concepts which are needed to provide the background which is necessary to extend the proofs of earlier work to this case. A critical issue is that of closure, i.e., defining the model such that any sequence of (systems, controls) has a convergent subsequence of the same type. One needs to introduce an extension of the Poisson measure (which serves a purpose analogous to that served by relaxed controls), which we call the relaxed Poisson measure, analogously to the use of the martingale measure concept given earlier to deal with controlled variance. The existence of an optimal control is a consequence of the development.