Reflecting Ito processes in a stochastic control problem

Let X(ë) be an Ito process with reflection at 0 and state space [0,•) and with nonanticipating infinitesimal coefficients μ(ë) and σ(ë). Let L^X(ë) be the process of local time at 0 for this X. Suppose that, for each t, (σ(t),μ(t)) are restricted to be in the set A(X(t)) where {A(y);0•y<•} is a given family of sets in R’+×R. Let Σ(x) be the class of all such Ito processes satisfying X(0)=x. Consider the stochastic control problem of maximizing P(L^X(Ta)•y•X(0)=x) over all X in Σ(x) where Ta=inf{t:X(t)=a}. It is shown here (under a natural hypothesis on the family A(ë)) that for all (a,y) in R’+×R’+ and all x in [0,a) the optimal solution is a reflecting diffusion which maximizes μ/σ².