Well-posedness of behavioral singular stochastic control problems

We investigate the well-posedness of a general class of singular stochastic control problems in which controls are processes of finite variation. We develop an abstract framework, which we then apply to storage management and portfolio investment problems under proportional transaction costs. Within this setting, we establish the existence of an optimal strategy in the class of randomized controls for a range of goal functionals, including cumulative prospect theory (CPT) preferences. To the best of our knowledge, this is the first treatment of behavioral storage management with the CPT goal functional. For the portfolio management problem, our analysis exploits the metrizable Meyer-Zheng topology to simplify proofs. We thoroughly investigate the applicability of the Skorokhod representation theorem for adapted random processes. Overall, the presented framework provides a clear separation between constraints imposed on the market model and properties required of the goal functional.