Optimal portfolio choice with path dependent benchmarked labor income: a mean field model. (English) Zbl 1489.34117
Summary: 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 [E. Biffis et al., SIAM J. Control Optim. 58, No. 4, 1906–1938 (2020; Zbl 1451.49031)]. Second, the labor income \(y_i\) of an agent \(i\) is benchmarked against the labor incomes of a population \(y^n := ( y_1 , y_2 , \dots , y_n )\) 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 \(n \to + \infty\) 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 E. Biffis et al. [loc. cit.; “Optimal portfolio choice with path dependent labor income: finite retirement time”, Preprint, arXiv:2101.09732].
MSC:
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
| 91G10 | Portfolio theory |
| 34K50 | Stochastic functional-differential equations |
| 49L12 | Hamilton-Jacobi equations in optimal control and differential games |
| 49L20 | Dynamic programming in optimal control and differential games |
Keywords:
life-cycle optimal portfolio with labor income following path dependent and law dependent dynamics; dynamic programming/optimal control of SDEs in infinite dimension with McKean-Vlasov dynamics and state constraints; second order Hamilton-Jacobi-Bellman equations in infinite dimension; verification theorems and optimal feedback controlsCitations:
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