  This paper extends the project initiated in arXiv:2002.00201 and studies a
lifecycle portfolio choice problem with borrowing constraints and finite
retirement time in which an agent receives labor income that adjusts to
financial market shocks in a path dependent way. The novelty here, with respect
to arXiv:2002.00201, is the fact that we have a finite retirement time, which
makes the model more realistic, but harder to solve. The presence of both
path-dependency, as in arXiv:2002.00201, and finite retirement, leads to a
two-stages infinite dimensional stochastic optimal control problem, a family of
problems which, to our knowledge, has not yet been treated in the literature.
We solve the problem completely, and find explicitly the optimal controls in
feedback form. This is possible because we are able to find an explicit
solution to the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB)
equation, even if state constraints are present. Note that, differently from
arXiv:2002.00201 , here the HJB equation is of parabolic type, hence the work
to identify the solutions and optimal feedbacks is more delicate, as it
involves, in particular, time-dependent state constraints, which, as far as we
know, have not yet been treated in the infinite dimensional literature. The
explicit solution allows us to study the properties of optimal strategies and
discuss their financial implications.
