Viability of an open set for stochastic control systems

The problem of compatibility of a stochastic control system and a set of constraints – the so called viability property – has been widely investigated during the last three decades. Given a stochastic control system, the question is to characterize sets A such that for any initial condition in A there exists a control ensuring that the associated stochastic process remains forever almost surely in A (this is called the viability property of A). When A is closed and the dynamics is continuous, the viability property has been characterized in the literature through several equivalent geometric conditions involving A, the drift and the diffusion of the control system. In this article we give a necessary and sufficient condition involving the boundary of an open set A ensuring the viability property of A, whenever A has a C2,1 boundary and the dynamics are Lipschitz. If moreover a classical convexity condition on the control dynamics holds true, we show that the viability of an open set A is equivalent to the viability of its closure. This last result is rather surprising, because several very elementary examples in the deterministic framework show that, in general, there is no such equivalence for a general open set A. We will also discuss examples illustrating that the above equivalence is wrong when either the boundary of A does not have enough regularity, or the dynamics are not Lipschitz continuous.