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

Boualem Djehiche, Fausto Gozzi, Giovanni Zanco, Margherita Zanella

Research output: Contribution to journalArticlepeer-review


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 yn≔(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).

Original languageEnglish (US)
Pages (from-to)48-85
Number of pages38
JournalStochastic Processes and their Applications
StatePublished - Mar 2022


  • Dynamic programming/optimal control of SDEs in infinite dimension with Mc Kean–Vlasov dynamics and state constraints
  • Life-cycle optimal portfolio with labor income following path dependent and law dependent dynamics
  • Second order Hamilton–Jacobi–Bellman equations in infinite dimension
  • Verification theorems and optimal feedback controls

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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