Switching games of stochastic differential systems

A two- player, zero- sum, switching game is formulated for general stochastic diff.erential systems and is studied using a combined dynamic programming and viscosity solution approach. The existence of the game value is proved. For the proof of the related dynamic programming principle ( DDP) for the lower and upper value functions, the measurability problem, of the same kind as mentioned in the paper of Fleming and Souganidis, is also encountered, and we are able to get around it via a delicate adaptation of their technique. Moreover, the traditional direct method to prove the time continuity of lower and upper value functions also gives rise to a serious measurability problem. To get around the new difficulty, a subtle dynamic programming argument is developed to obtain the time continuity, which in return is used to derive the DDP for random intermediate times from the DDP with deterministic intermediate times.