Stochastic optimal control : theory and its application to the optimal investment-consumption problem with proportional transaction costs

Abstract

The aim of this dissertation is to study the theory of classical and singular stochastic optimal control at a rigorous level and to apply such theory to the optimal investment-consumption problem with proportional transaction costs. The behaviour of a stochastic dynamical system may be affected by various control processes. Stochastic optimal control deals with selecting that particular process which optimises some performance measure. Classical stochastic optimal control problems involve those problems in which the displacement of the state process is continuous. Singular stochastic control problems are similar to classical stochastic control problems except for allowing the displacement of the state to be possibly discontinuous. In this dissertation, the dynamical systems considered are represented by Markov diffusion processes that are solutions to stochastic differential equations, while the performance measure is given in additive form. The Method of Dynamic Programming will then be used as a tool to find the stochastic optimal control. This will lead to solving a partial differential equation. Having introduced the general theory, the optimal investment-consumption problem with proportional transaction costs will be then discussed extensively. This problem aims to maximise the total expected utility of consumption by selecting optimally the distribution of wealth over various investment opportunities and by consuming optimally, in presence of transaction costs. The transaction costs to be considered are proportional to the amount transacted. This will lead to a free-boundary problem, the solution of which is quite arduous to obtain. Hence an approximate solution to such problem will be found numerically using the Moving Boundary Method, which transforms the free-boundary problem into a sequence of fixed-boundary problems.