Stochastic viability for compact sets in terms of the distance function (Q2753270)

The authors consider the multidimensional stochastic differential equation NEWLINE\[NEWLINE dX(t,x)=b(X(t,x)) dt+\sigma(X(t,x)) dW(t),\quad X(0,x)=x, NEWLINE\]NEWLINE with \(m\)-dimensional Brownian motion \(W\) and continuous coefficients \(b:R^n\to R^n\) and \(\sigma:R^m\to R^n\). A set \(K\subset R^n\) is said to be viable for this equation if, for every \(x\in K\), there exists a weak solution \(X(t,x)\) such that \(X(t,x)\in K\) for all \(t\geq 0\). Under certain smoothness hypothesis on the distance function \(\varphi_K(x):=\inf_{y\in K}|x-y|^2\), \(x\in K\) (which is always satisfied for convex \(K\)), it is proved that \(K\) is viable if and only if \(L_K\varphi(x)\leq 0\) for almost all \(x\) in a neighborhood of \(K\), where NEWLINE\[NEWLINE L_Kf(x):=\tfrac 12\operatorname{Tr}[D^2f(x)\sigma(\pi_Kx)\sigma^*(\pi_Kx)] +\langle Df(x),b(\pi_Kx)\rangle. NEWLINE\]NEWLINE Here \(\pi_Kx\) denotes the projection of \(x\) onto \(K\).