PUB - Publikationen an der Universität Bielefeld

Stochastic singular control models (such as optimization problems, games and mean field games) refer to a class of problems in which some agents want to optimize a certain performance criterion by acting in a random environment which evolves in continuous time, and in which the effect of the agents' action on both the environment and on the performance is (linearly) proportional to the size of the action. Applications include investment and portfolio selection in finance, inventory management in operations research, control of queueing networks, dividend and equity issuance in insurance mathematics, spacecraft control in aerospace engineering, or rational harvesting in mathematical biology. Models involving stochastic singular controls raise many unsolved mathematical issues which represent a relevant limitation to their theoretical understanding. In this thesis, we provide mathematical tools and structural conditions which allow to address problems of existence, characterization and approximation of solutions in optimization problems and games involving stochastic singular controls. For a class of optimal stochastic control problems with singular controls, we characterize the optimal control as the unique solution to a related Skorokhod reflection problem. We prove that the optimal control only acts when the underlying diffusion attempts to exit the so-called waiting region, and that the direction of this action is prescribed by the derivative of the value function. We next consider problems concerning existence and approximation of Nash equilibria in N-player stochastic games of multi-dimensional singular control with submodular costs. In a not necessarily Markovian setting, we establish the existence of Nash equilibria via Tarski's fixed point theorem, and we propose an algorithm to determine a Nash equilibrium. Moreover, we derive relations between weak (distributional) Nash equilibria of the game of singular control and the Nash equilibria of stochastic games with regular controls. Further, we study mean field games with regular and singular controls and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Also, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist minimal and maximal solutions. Finally, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. Our approach also allows to prove existence of a strong solution for a class of submodular mean field games with common noise, where the representative player at equilibrium interacts with the (conditional) mean of its state's distribution. Finally, we analyse stationary mean field games with singular controls in which the representative player interacts with a long-time weighted average of the population through a discounted and an ergodic performance criterion. This class of games finds natural applications in the context of optimal productivity expansion in dynamic oligopolies. We prove existence and uniqueness of the mean field equilibria, which are completely characterized through nonlinear equations. Furthermore, we relate the mean field equilibria of the discounted and the ergodic games by showing the validity of an Abelian limit. The latter also allows to approximate Nash equilibria of symmetric N-player ergodic singular control games through the mean field equilibrium of the discounted game.