A problem of finite-horizon optimal switching and stochastic control for utility maximisation
摘要
In this paper, we investigate the utility maximisation problem faced by an economic agent who can switch jobs under a mandatory retirement date. The agent must consider not only optimal consumption and investment, but also the optimal job-switching decision. Therefore the utility maximisation problem incorporates both optimal switching and stochastic control features over a finite horizon. To address this challenge, we employ a dual martingale approach and derive the dual problem as a finite-horizon pure optimal switching problem. We then apply the theory of a double obstacle problem, using non-standard arguments to examine the analytical properties of the resulting system of parabolic variational inequalities, including its two free boundaries. Building on these analytical properties, we establish a duality theorem and characterise the optimal job-switching strategy through time-varying wealth boundaries. Furthermore, we derive integral equation representations satisfied by the optimal strategies and present numerical results based on these representations.