Invariance of stochastic control systems with deterministic arguments

We prove that a closed set K of a finite-dimensional space is invariant under the stochastic control system dX=b(X,v(t))dt+σ(X,v(t))dW(t),v(t)∈U, if and only if it is invariant under the deterministic control system with two controls x′=b(x,v(t))−12∑j=1mDσj(x,v(t))σj(x,v(t))+σ(x,v(t))u(t),u(t)∈H1,v(t)∈U. This extends the well-known result of stochastic differential equations to stochastic control systems. Furthermore, we ask only C1,1 regularity of the diffusion σ instead of the usual assumption σ∈C2. In this way our result is new even for stochastic differential equations. The arguments of the proof are based on estimates between solutions of the stochastic control system with time independent controls and families of solutions {xω(·)}ω∈Ω to the deterministic control system x′=σ(x,vω)uω(t),uω(t)∈H1 with appropriately chosen controls uω(t) and vω∈U.