On time-inconsistent investment and dividend problems

In this thesis, we reconsider some continuous-time stochastic control problems in finance and insurance. To incorporate some well-documented behavioural features of human beings, we consider the situation where the discounting is non-exponential. This situation is far from trivial and renders the optimisation problem to be a non-standard one, namely, a time-inconsistent stochastic control problem since Bellman’s principle of optimality does not hold. In this situation, the optimal control is time-inconsistent, namely, a strategy that is optimal for the initial time may not be optimal later. Three self-contained papers are included in this thesis, each of which is concerned with one specific optimisation problem. We analyse these problems within a game theoretic framework and try to find the time-consistent equilibrium strategies for each problem. In Chapter 2, we study the dividend maximisation problem in a diffusion risk model and try to find an equilibrium strategy within the class of feedback controls. We assume that the dividends can only be paid at a bounded rate and consider the ruin risk in the dividend problem. We obtain an equilibrium HJB equation and verification theorem for a general discount function, and get closed-form solutions in two examples. In Chapter 3, we investigate the defined benefit pension problem, where the aim of the decision-maker is to minimise two types of risks: the contribution rate risk and the solvency risk, by considering a quadratic performance criterion. In our model, we assume that the benefit outgo is constant and the pension fund can be invested in a riskfree asset and a risky asset whose return follows a geometric Brownian motion. We characterise the time-consistent equilibrium strategy and value function in terms of the solution of a system of integral equations. The existence and uniqueness of the solution is verified and the approximation of the solution is obtained. In Chapter 4, we consider the consumption-investment problem with logarithmic utility in a non-Markovian framework. The coefficients in our model are assumed to be adapted stochastic processes. We first study an N-person differential game and adopt a martingale method to solve an optimisation problem of each player and characterise their optimal strategies and value functions in terms of the unique solutions of BSDEs. Then by taking the limit, we show that a time-consistent equilibrium consumption-investment strategy of the original problem consists of a deterministic function and the ratio of the market price of risk to the volatility, and the corresponding equilibrium value function can be characterised by the unique solution of a family of BSDEs parameterised by a time variable.