Jump-Di usions With Controlled Jumps: Existence and Numerical Methods

A comprehensive development of eeective numerical methods for stochas-tic control problems in continuous time, for reeected jump-diiusion models , is given in 10, 11, 16]. 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 aaects 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 10, 11, 16] to this case. A critical issue is that of closure: i.e., deening 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 in 11] to deal with controlled variance. The existence of an optimal control is a consequence of the development.