Topics On Optimal Portfolio Strategies And Dual Risk Model

With the growing integration of world capital markets, great emphasis has been placed on exploring portfolio selection strategies. This dissertation studies kinds of optimal portfolio selection problems with constraints and dual risk model, which are very meaningful and comprehensively applicable in portfolio management. Meanwhile the optimal portfolio selection strategies are supplied for the different kinds of investors.In Chapter 2, we study a general portfolio selection problem for any utility function where all market coefficients are random and the wealth process is not allowed to be below a benchmark wealth process. The problem is completely solved by using a decomposition approach. After a system of equations for a Lagrange multiplier is solved, the portfolio selection problem are derived as the replicating portfolios of contingent claims. And some Monte-Carlo simulations are done to show the effect of the strategy.Own to the investors'investment objectives could be various, we especially study in Chapter 3 the portfolio selection problem of tracking the target wealth process with optimizing consumption, in which the individual preference is consid-ered. By the Hamilton-Jacobi-Bellman approach, we obtain the exact formula of the strategy for this model. Meanwhile, the sensitivity of this strategy to the individual preference has been analyzed.Furthermore, we consider the optimal portfolio on tracking the expected wealth process with liquidity constraints in Chapter 4. The constrained optimal portfolio is first formulated as minimizing the cumulate variance between the wealth process and the expected wealth process. Then the dynamic programming methodology is applied to reduce this problem to solving the Hamilton-Jacobi-Bellman equation coupled with the liquidity constraint, and the method of Lagrange multiplier is applied to handle the constraint. We also propose a numerical method to solve the constrained HJB equation and the constrained optimal portfolio.In Chapter 5, we also study the optimal constant barrier strategies in a Markov-modulated dual risk model, in which the gain arrivals, gain sizes and expenses are influenced by a Markov process. Integro-differential equations with boundary conditions for the expected value of the discounted dividends until ruin are derived. In the case of exponentially distributed gain sizes and two state Markov process, the equations are solved and the optimal barrier is obtained via numerical approach.