Reflecting Ito Processes in a Stochastic Control Problem

Let X(·) be an Ito process with reflection at 0 and state space [0, ∝) and with nonanticipating infinitesimal coefficients μ(·) and σ(·). Let L^X(·) 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(T_a) ≤ y|X(0) = x) over all X in Σ(x) where T_a = inf{t : X(t) = a}. It is shown here (under a natural hypothesis on the family A(·)) 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.