Limitations of dynamic programming approach: singularity and time inconsistency

Two failures of the dynamic programming (DP) approach to the stochastic optimal control problem are investigated. The first failure arises when we wish to solve a class of certain singular stochastic control problems in continuous time. It has been shown by Lasry and Lions (2000) that this difficulty can be overcome by introducing equivalent standard stochastic control problems. To solve this class of singular stochastic control problems, it remains to solve the equivalent standard stochastic control problems. Since standard stochastic control problems can be solved by applying the DP approach, this then solves the first failure. In the first part of the thesis, we clarify the idea of Lasry and Lions and extend their work to the case of controlled processes with jumps. This is particularly important in financial modelling where such processes are widely applied. For the purpose of application, we applied our result to an optimal trade execution problem studied by Lasry and Lions (2007b). The second failure of the DP approach arises when we wish to solve a multiperiod portfolio selection problem in which a mean-standard-deviation type criterion (a non-separable criterion) is used. We formulate such a problem as a discrete time stochastic control problem. By adapting a pseudo dynamic programming principle, we obtain a closed form optimal strategy for investors whose risk tolerances are larger than a lower bound. As a consequence, we develop a multiperiod portfolio selection scheme. The analysis is performed in the market of risky assets only, however, we allow both market transitions and intermediate cash injections and offtakes. This work provides a good basis for future studies of portfolio selection problems with selection criteria chosen from the class of translation-invariant and positive-homogeneous risk measures