Abstract :
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 ofŽsystems, controls.has a convergent subsequence of the same type.
One needs to introduce an extension of the Poisson measure Žwhich 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
