Stochastic control problems in optimal consumption and optimal dividends

Abstract

This thesis studies two applications of stochastic control in quantitative finance and insurance, namely one problem of optimal entry decision making and dynamic consumption with habit formation and one problem of optimal dividend payment for an insurance group in face of external default risk. By using dynamic programming argument and some delicate partial differential equation (PDE) analysis, we can characterize the value function of each control problem as the solution to the associated Hamilton-Jacobi-Bellman (HJB) variational inequality in a classical sense or in a viscosity sense. In the first project, we consider a composite problem to choose an optimal entry time from complete market information to incomplete information bearing information costs. Starting from the chosen time, the investor no longer pays the fee for acquiring the extra market information and chooses dynamic investment and consumption strategies through partial observations of the public stock price. In addition, the habit formation preference is considered for the dynamic consumption problem. By employing the stochastic Perron's method, the value function of this composite problem is proved to be a viscosity solution of the HJB variational inequality. For the interior optimal investment-consumption problem, the feedback control policies are obtained. The numerical illustration of the continuation region and stopping region is also presented. In the second project, a multi-dimensional optimal dividend problem for an insurance group is formulated and studied. The novelty of our work is to incorporate the systemic risk modelled by the contagious credit default risk among subsidiaries. That is, each subsidiary of the insurance group runs a product line and all subsidiaries suffer from the external credit risk from the financial market. The default contagion is considered in the sense that one default event may increase the default probabilities of all surviving subsidiaries. By studying the recursive system of the Hamilton-Jacobi-Bellman variational inequalities (HJBVIs), the optimal singular dividend of each subsidiary satisfies a barrier type and the optimal barrier is dynamically modulated by the current default state. In the case of two subsidiaries, the value function and optimal barriers are given in analytical forms, allowing us to conclude that the optimal barrier of one subsidiary decreases if the other subsidiary defaults.