Consumption-Investment Problem in Rank-Based Models
David Itkin
TL;DR
This work analyzes a consumption-investment problem in a high-dimensional market with rank-based asset dynamics, deriving an HJB equation with Neumann boundary conditions and proving a verification theorem that links the value function to a ranked-state representation. By exploiting a rank-based ansatz, the authors reduce the problem to a reflected diffusion on the ranked domain and obtain explicit, Merton-type solutions in first-order models. They show that unconstrained, open-market, and fully invested scenarios admit closed-form optimal strategies tied to rank-based Merton fractions, with open-market constraints yielding particularly tractable explicit formulas not readily available in standard models. The results provide practical, rank-aware strategies and facilitate calibration via collision estimators, offering new insights into portfolio choice when only rank information drives asset dynamics.
Abstract
We study a consumption-investment problem in a multi-asset market where the returns follow a generic rank-based model. Our main result derives an HJB equation with Neumann boundary conditions for the value function and proves a corresponding verification theorem. The control problem is nonstandard due to the discontinuous nature of the coefficients in rank-based models, requiring a bespoke approach of independent mathematical interest. The special case of first-order models, prescribing constant drift and diffusion coefficients for the ranked returns, admits explicit solutions when the investor is either (a) unconstrained, (b) abides by open market constraints or (c) is fully invested in the market. The explicit optimal strategies in all cases are related to the celebrated solution to Merton's problem, despite the intractability of constraint (b) in that setting.