Stochastic control and compatible subsets of constraints

Given a stochastic differential control system and a closed set K in Rn, we study the that, with probability one, the associated solution of the control system remains for ever in the set K. This set is called the viability kernel of K. If N is equal to the whole set K, K is said to be viable. We prove that, in the general case, the viability kernel itself is viable and we characterize it through some partial differential equations. We prove that, under suitable assumptions, also the boundary of N is viable. As an application, we give a new characterization of the value function of some optimal control problem.