  We present a robust version of the life-cycle optimal portfolio choice
problem in the presence of labor income, as introduced in Biffis, Gozzi and
Prosdocimi ("Optimal portfolio choice with path dependent labor income: the
infinite horizon case", SIAM Journal on Control and Optimization, 58(4),
1906-1938.) and Dybvig and Liu ("Lifetime consumption and investment:
retirement and constrained borrowing", Journal of Economic Theory, 145, pp.
885-907). In particular, in Biffis, Gozzi and Prosdocimi the influence of past
wages on the future ones is modelled linearly in the evolution equation of
labor income, through a given weight function. The optimisation relies on the
resolution of an infinite dimensional HJB equation. We improve the state of art
in three ways. First, we allow the weight to be a Radon measure. This
accommodates for more realistic weighting of the sticky wages, like, e.g., on a
discrete temporal grid according to some periodic income. Second, there is a
general correlation structure between labor income and stocks market. This
naturally affects the optimal hedging demand, which may increase or decrease
according to the correlation sign. Third, we allow the weight to change with
time, possibly lacking perfect identification. The uncertainty is specified by
a given set of Radon measures $K$, in which the weight process takes values.
This renders the inevitable uncertainty on how the past affects the future, and
includes the standard case of error bounds on a specific estimate for the
weight. Under uncertainty averse preferences, the decision maker takes a maxmin
approach to the problem. Our analysis confirms the intuition: in the infinite
dimensional setting, the optimal policy remains the best investment strategy
under the worst case weight.
