Consumption-Investment Problem in Rank-Based Models

TL;DR

This paper addresses a complex consumption-investment optimization in multi-asset markets with rank-based return models, deriving new mathematical results including an HJB equation and explicit solutions in special cases.

Contribution

It introduces a novel approach to solving a nonstandard control problem with rank-based models, including explicit solutions for certain simplified cases.

Findings

Derived an HJB equation with Neumann boundary conditions.

Provided explicit solutions for first-order models under various constraints.

Connected optimal strategies to Merton's classical problem.

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.