Abstract
In this paper we propose a duality approach to solving an optimal consumption-portfolio selection problem in which an economic agent can choose (irreversible) the preference change time in finite horizon. We assume that after preference change, the agent's coefficient of relative risk aversion becomes higher than before. We use the martingale method and study the variational inequality or free boundary problem arising from the dual problem. Based on partial differential equation (PDE) theory, we analytically characterize the variational inequality and establish the duality relationship. Specifically, we prove that the agent will change his/her preference as soon as the agent's wealth reaches the threshold of wealth as a function of time.
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