Forcing a Stochastic Process to Stay in or to Leave a Given Region

Systems defined by dx(t)=a[x(t),t]dt+B[x(t),t]u(t)dt+N1/2[x(t),t]dW(t), where x(t) is the state variable, u(t) is the control variable, a is a vector function, B and N are matrices and W(t) is a Brownian motion process, are considered. The aim is to minimize the expected value of a cost function with quadratic control costs on the way and terminal cost function K(T), where T=inf{s:x(s).D.x(t)=x},D being a given region in Rn. The function K is taken to be 0 if T.(.)., where . is a positive constant and +. if T<(>). when the aim is to force x(t) to stay in (resp., to leave) the region C, the complement of D. A particular one-dimensional problem is solved explicitly and a risk-sensitive version of the general problem is also considered.