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 sol-vency 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 approxi-mation 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 strate-gies 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.