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This thesis addresses three optimisation problems. The first problem concerns static portfolio optimization. Empirical evidence suggests that asset dynamics can be characterised by a multifactor representation of asset returns. In this context, investors can capture the premium or hedge the risk associated with a particular factor through factor-mimicking portfolios. We examine different methods for constructing optimal factor-mimicking portfolios. We provide analytical considerations in the construction of factor-mimicking portfolios, along with empirical evidence. Also, we illustrate potential practical applications of factor-mimicking portfolios in the institutional investment process. The second problem addresses continuous time optimal consumption and investment for an agent investing in a complete arbitrage-free market consisting of several risky assets and a bank account. In particular, we consider a general model in which the agent's preferences exhibit a linear habit formation pattern that captures the effect of past consumption on current utility. Using duality methods, we establish the existence of an optimal consumption and investment strategy. We also prove that the optimal portfolio consumption pair can be expressed in terms of the solution to the Hamilton-Jacobi-Bellman equation that the problem's value function satisfies. The third problem concerns a general one-dimensional lro diffusion which must be maintained within an externally specified bounded interval by means of an impulse control process. We minimise a long-term average criterion that penalises deviations of the state process from a given nominal point within this region as well as the use of impulsive control effort. We solve the resulting optimisation problem and we provide an explicit optimal control strategy under general assumptions. The model that we study is motivated by several applications, including the problem of determining an optimal central bank intervention strategy aiming at the control of an exchange rate or an inflation rate, as well as the problem of designing optimal contribution policies in defined benefit pension plans. |
