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Consumption-Investment Problem in Rank-Based Models

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  • David Itkin

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.

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  • David Itkin, 2025. "Consumption-Investment Problem in Rank-Based Models," Papers 2510.20763, arXiv.org.
    • Handle: RePEc:arx:papers:2510.20763
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    References listed on IDEAS

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    1. David Itkin & Martin Larsson, 2024. "Calibrated rank volatility stabilized models for large equity markets," Papers 2403.04674, arXiv.org.
    2. Banner, Adrian D. & Ghomrasni, Raouf, 2008. "Local times of ranked continuous semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1244-1253, July.
    3. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    4. David Itkin & Martin Larsson, 2021. "Open Markets and Hybrid Jacobi Processes," Papers 2110.14046, arXiv.org, revised Mar 2024.
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