Abstract:  Imagine that two players observe the evolution of a random motion, to be modeled in this talk as a diffusion process. One of the players (the controller) can intervene and influence the local characteristics, or "dynamics", of the process, such as its local drift and variance; whereas the other player (the stopper) can stop the game at any time of his choice. When the game ends the controller pays the stopper a certain amount, which depends on the position of the process at the time of termination - and perhaps also on the entire trajectory of the process up to that time. It is, of course, in the interest of the controller (respectively, of the stopper) to minimize (respectively, to maximize) this amount, at least in expectation.

Does the resulting stochastic zero-sum game have a value? Does it have a saddle point (that is, a pair of strategies by the two players that are best responses to each other)? If so, how does one characterize, or even compute, these quantities? How about the corresponding non-zero-sum game, in which one looks for Nash equilibria rather than for saddles? Questions such as these have been asked and addressed only recently, and very few results exist. We shall describe some of them, small islands in a vast sea of open questions.

(Joint work with William Sudderth, University of Minnesota.)