Time inconsistent portfolio selection and indifference pricing

In this thesis, we study three time inconsistent portfolio selection and option pricing problems with regime switching via the equilibrium approach. We first completely solve a type of time inconsistent utility maximisation problems. We obtain the equilibrium policy laws for both the time inconsistent CARA and CRRA utility maximisation problems and point out that, for the CARA utility maximisation problem, the equilibrium policy law coincides with the naive policy law. We next study a time inconsistent option pricing problem in an incomplete market via the indifference pricing approach. We provide both the equilibrium utility and mean-variance indifference pricing rule. For the equilibrium utility indifference pricing rule, we give the PDE system characterising the equilibrium utility indifference price and show the local existence and uniqueness of this PDE system in a simplified model; for the equilibrium mean-variance indifference pricing rule, we give the explicit formula for the equilibrium mean-variance indifference price. We also discuss the marginal equilibrium indifference prices and some properties of the equilibrium pricing rules.