The Lagrange and the vanishing discount techniques to controlled diffusions with cost constraints

In this paper we introduce two useful methods to compute optimal control policies for either the discounted or the average payoff criterion with cost constraints when the dynamic system evolves as a n-dimensional diffusion processes. As for the attribute “cost constraints” we mean the coexistence of a given cost function that in general is dominated above by another function (in particular by a constant) playing the role of an extra constraint in the control model. To deduce optimality results for the discounted case, we employ the Lagrange multipliers technique along with dynamic programming arguments. Then, the vanishing discount method is applied to easily obtain average optimal policies. We support our theory by providing an example on pollution accumulation problem.