On the infinite time solution to state-constrained stochastic optimal control problems

A method is presented for solving the infinite time Hamilton-Jacobi-Bellman (HJB) equation for certain state-constrained stochastic problems. The HJB equation is reformulated as an eigenvalue problem, such that the principal eigenvalue corresponds to the expected cost per unit time, and the corresponding eigenfunction gives the value function (up to an additive constant) for the optimal control policy. The eigenvalue problem is linear and hence there are fast numerical methods available for finding the solution.