Robust portfolio choice with sticky wages

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 [SIAM J. Control Optim., 58 (2020), pp. 1906--1938] and Dybvig and Liu [J. Econom. Theory, 145 (2010), pp. 885--907]. In particular, in Biffis, Gozzi, and Prosdocimi the influence of past wages on the future ones is modeled linearly in the evolution equation of labor income, through a given weight function. The optimization 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, e.g., on a discrete temporal grid according to some periodic income. Second, there is a general correlation structure between labor income and stock markets. 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.