Optimal portfolio choice with path dependent benchmarked labor income: A mean field model

We consider the life-cycle optimal portfolio choice problem faced by an agent receiving labor income and allocating her wealth to risky assets and a riskless bond subject to a borrowing constraint. In this paper, to reflect a realistic economic setting, we propose a model where the dynamics of the labor income has two main features. First, labor income adjusts slowly to financial market shocks, a feature already considered in Biffis et al. (2015). Second, the labor income yi of an agent i is benchmarked against the labor incomes of a population y^n≔(y1,y2,…,yn) of n agents with comparable tasks and/or ranks. This last feature has not been considered yet in the literature and is faced taking the limit when n→+∞ so that the problem falls into the family of optimal control of infinite-dimensional McKean–Vlasov Dynamics, which is a completely new and challenging research field. We study the problem in a simplified case where, adding a suitable new variable, we are able to find explicitly the solution of the associated HJB equation and find the optimal feedback controls. The techniques are a careful and nontrivial extension of the ones introduced in the previous papers of Biffis et al. (2015, 0000).