  We study an agent's lifecycle portfolio choice problem with stochastic labor
income, borrowing constraints and a finite retirement date. Similarly to
arXiv:2002.00201, wages evolve in a path-dependent way, but the presence of a
finite retirement time leads to a novel, two-stage infinite dimensional
stochastic optimal control problem with explicit optimal controls in feedback
form. This is possible as we find an explicit solution to the associated
Hamilton-Jacobi-Bellman (HJB) equation, which is an infinite dimensional PDE of
parabolic type. The identification of the optimal feedbacks is delicate due to
the presence of time-dependent state constraints, which appear to be new in the
infinite dimensional stochastic control literature. The explicit solution
allows us to study the properties of optimal strategies and discuss their
implications for portfolio choice. As opposed to models with Markovian
dynamics, path dependency can now modulate the hedging demand arising from the
implicit holding of risky assets in human capital, leading to richer asset
allocation predictions consistent with wage rigidity and the agents learning
about their earning potential.
