Singular Control of Optional Random Measures

In this thesis, we study the problem of maximizing certain concave functionals on the space of optional random measures. Such functionals arise in microeconomic theory where their maximization corresponds to finding the optimal consumption plan of some economic agent. As an alternative to the well–known methods of Dynamic Programming, we develop a new approach which allows us to clarify the structure of maximizing measures in a general stochastic setting extending beyond the usually required Markovian framework. Our approach is based on an infinite–dimensional version of the Kuhn–Tucker Theorem. The implied first–order conditions allow us to reduce the maximization problem to a new type of representation problem for optional processes which serves as a non–Markovian substitute for the Hamilton–Jacobi–Bellman equation of Dynamic Programming. In order to solve this representation problem in the deterministic case, we introduce a time–inhomogeneous generalization of convexity. The stochastic case is solved by using an intimate relation to the theory of Gittins–indices in optimal dynamic scheduling. Closed–form solutions are derived under appropriate conditions. Depending on the underlying stochastics, maximizing random measures can be absolutely continuous, discrete, and also singular. In the microeconomic context, it is natural to embed the above maximization problem in an equilibrium framework. In the last part of this thesis, we give a general existence result for such an equilibrium.