Applications of Stochastic Control to Portfolio Selection Problems

Portfolio selection is an important problem both in academia and in practice. Due to its significance, it has received great attention and facilitated a large amount of research. This thesis is devoted to structuring optimal portfolios using different criteria. Participating contracts are popular insurance policies, in which the payoff to a policyholder is linked to the performance of a portfolio managed by the insurer. In Chapter 2, we consider the portfolio selection problem of an insurer that offers participating contracts and has an S-shaped utility function. Applying the martingale approach, closed-form solutions are obtained. The resulting optimal strategies are compared with two portfolio insurance hedging strategies, e.g. Constant Proportion Portfolio Insurance strategy and Option Based Portfolio Insurance strategy. We also study numerical solutions of the portfolio selection problem with constraints on the portfolio weights. In Chapter 3, we consider the portfolio selection problem of maximizing a performance measure in a continuous-time diffusion model. The performance measure is the ratio of the overperformance to the underperformance of a portfolio relative to a benchmark. Following a strategy from fractional programming, we analyze the problem by solving a family of related problems, where the objective functions are the numerator of the original problem minus the denominator multiplied by a penalty parameter. These auxiliary problems can be solved using the martingale method for stochastic control. The existence of a solution is discussed in a general setting and explicit solutions are derived when both the reward and the penalty functions are power functions. In Chapter 4, we consider the mean-risk portfolio selection problem of optimizing the expectile risk measure in a continuous-time diffusion model. Due to the lack of an explicit form for expectiles and the close relationship with the Omega measure, we propose an alternative optimization problem with the Omega measure as an objective and show the equivalence between the two problems. After showing the solution for the mean-expectile problem is not attainable but the value function is finite, we modify the problem with an upper bound constraint imposed on the terminal wealth and obtain the solution via the Lagrangian duality method and pointwise optimization technique. The global expectile minimizing portfolio and efficient frontier are also considered in our analysis. In Chapter 5, we consider the utility-based portfolio selection problem in a continuous-time setting. We assume the market price of risk depends on a stochastic factor that satisfies an affine-form, square-root, Markovian model. This financial market framework includes the classical geometric Brownian motion, the constant elasticity of variance (CEV) model and the Heston's model as special cases. Adopting the Backward Stochastic Differential Equation (BSDE) approach, we obtain the closed-form solutions for power, logarithm, or exponential utility functions, respectively. Concluding remarks and several potential topics for further research are presented in Chapter 6