PUB - Publikationen an der Universität Bielefeld

We propose a tractable dynamic framework for the joint determination of optimal con- sumption, portfolio choice, and healthcare irreversible investment. Our model is based on a Merton’s portfolio and consumption problem, where, in addition, the agent can choose the time at which un- dertaking a costly lump sum health investment decision. Health depreciates with age and directly affects the agent’s mortality force, so that investment into healthcare reduces the agent’s mortality risk. The resulting optimization problem is formulated as a stochastic control-stopping problem with a random time-horizon and state-variables given by the agent’s wealth and health capital. We trans- form this problem into its dual version, which is now a two-dimensional optimal stopping problem with interconnected dynamics and finite time-horizon. Regularity of the optimal stopping value func- tion is derived and the related free boundary surface is proved to be Lipschitz continuous and it is characterized as the unique solution to a nonlinear integral equation. In the original coordinates, the agent thus invests into healthcare whenever her wealth exceeds an age- and health-dependent transformed version of the optimal stopping boundary.

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MSC Classification: 91B70, 93E20, 60G40