Two Types Of Portfolio Management Problems And Martingale Methods

In this thesis,we investigate two important portfolio management problems under the framework of the expected utility theory: a pension fund management problem with liquidation constraints and mixed incentive schemes;a robust consumption-investment problem with time-varying confidence sets.For the pension fund management problem with liquidation constraints and mixed incentive schemes,we consider a non-linear payoff function and a liquidation constraint to describe performance-based incentive contracts and financial security requirements,respectively.Therefore,the management problem is a utility maximization problem with a non-concave and non-differentiable utility function and an infimum constraint.Because of these two innovations,the envelope of the utility function depends on the contract parameters and can be divided into three categories.Complex categories,the non-concavity,and the non-differentiability both cause difficulties in obtaining explicit solutions.Fortunately,studying general piecewise hyperbolic absolute risk aversion(HARA)utilities,we explicitly obtain a unified form of the optimal strategies and the optimal fund wealth processes for all three different categories by the martingale method.Based on these explicit expressions,we illustrate the advantages of mixed incentive contracts by comparing the risk characteristics of incentives;we find that mixed incentive contracts can achieve Pareto improvement by comparing the expected utilities of the manager and the investor.For the robust consumption-investment problem with time-varying confidence sets,we propose time-varying confidence sets in the robust model for characterizing the robustness and time dependence of the prior information.To the best of our knowledge,there is no “one-size-fits-all” approach to solve a robust optimization problem under the jump-diffusion model.Therefore,we systematically propose a robust martingale method for solving such problems,that is,proposing a deterministic functional(global kernel)and using its saddle point to construct a solution of the robust optimization problem.In practice,it is very difficult to find the saddle point because the domain of the global kernel is in an infinite dimensional space,which consists of measurable functions.Specifically,we not only need to demonstrate the measurability of the saddle point,but also need to verify a minimax property.To this end we propose and prove a measurable saddle point theorem and semi-explicitly find a saddle point of the global kernel by the variational method and the dynamic programming method.Based on the semi-explicit solution of the optimal strategy and the worst-case model,we reveal the rule of choosing the worst-case model and explain the impact of the model uncertainty on the optimal strategy,under a simple robust jump-diffusion model.