Singular stochastic control in risk-sensitive optimisation and equilibrium asset pricing with proportional transaction costs

In this thesis, we investigate the applications of singular stochastic control in two optimization problems. In the first part, we consider a two-sided singular stochastic control problem with a risk-sensitive ergodic criterion. In particular, we consider a stochastic system whose uncontrolled dynamics are modelled by a linear diffusion. The control that can be applied to this system is modelled by an additive finite variation process. The objective of the control problem is to minimise a risk-sensitive long-term average criterion that penalises deviations of the controlled process from a given interval as well as the expenditure of control effort. We derive the complete solution to the problem under general assumptions by deriving a C² solution to its HJB equation. To this end, we use the solutions to a suitable family of Sturm-Liouville eigenvalue problems. In the second part of this thesis, we study a risk-sharing equilibrium with proportional transaction costs. We consider an economy with two agents, each of whom receive a cumulative endowment flow which is modelled as a stochastic integral of a deterministic continuous function of the economy’s state, which is modelled by means of a general Itô diffusion. Each of the two heterogeneous agents have mean-variance preferences and can also trade a risky asset to hedge against the fluctuations of their endowment streams. We determine the agents’ optimal (Radner) equilibrium trading strategies in the presence of proportional transaction costs. In particular, we derive a new free-boundary problem that provides the solution to the agents’ optimisation problem in equilibrium. Furthermore, we derive the explicit solution to this free-boundary problem when the problem data is such that the frictionless optimiser is a strictly increasing or a strictly increasing and then strictly decreasing function of the economy’s state. Finally, we derive small transaction cost asymptotics.