Portfolio with consumption and terminal gains under loss aversion

A continuous-time portfolio selection problem with consumption and terminal gains was considered in the framework of prospect theory. The Inada conditions for the utility functions were discarded by assuming a regularity condition on the terminal utility. First, the problem with the reference point depending on the wealth was considered, and the corresponding Hamilton-Jacobi-Bellman (HJB) equation was derived. Then, by assuming that the terminal utility relies on the gains process, a new model with the reference point as part of the control was established. This makes the optimal control problem non-Markovian. To deal with this problem, the idea for transforming the Asian option pricing problem into a Markov problem was used. A singular Markov control problem was yielded, and then acorresponding HJB variational inequality was derived.