Utility Maximisation: Non-concave utility and non linear expectation

Since the birth of mathematical nance, portfolio selection has been one of the topics which have attracted a lot of interest, with models formulated in discrete and continuous time and developed in complete and incomplete markets. In conventional or neoclassical finance, many models are based off the assumption that agents make decisions by maximising their expected utility. Deviations between models and market observations have generated a recent field of study, behavioural finance, which incorporates psychology, sociology and finance together to resolve observed phenomenon like bubbles which conventional finance cannot explain. In this thesis, we will be restricting ourselves to the complete continuous market and look at a new formulation of expected utility maximisation with behavioural finance elements incorporated into it, namely S-shaped utilities and probability distortions. We consider the three general cases of expected utility maximisation: utility from terminal wealth, utility from consumption and utility from terminal wealth and consumption. We shall review the neoclassical problems and then explore the cases with behavioural elements installed. Key Words: Portfolio Selection, continuous time, martingale approach, Sshaped function, probability distortion, cumulative prospect theor