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Optimal investment in a large population of competitive and heterogeneous agents

Author

Listed:
  • Ludovic Tangpi

    (Princeton University)

  • Xuchen Zhou

    (Princeton University)

Abstract

This paper studies a stochastic utility maximisation game under relative performance concerns in finite- and infinite-agent settings, where a continuum of agents interact through a graphon (see definition below). We consider an incomplete market model in which agents have CARA utilities, and we obtain characterisations of Nash equilibria in both the finite-agent and graphon paradigms. Under modest assumptions on the denseness of the interaction graph among the agents, we establish convergence results for the Nash equilibria and optimal utilities of the finite-player problem to the infinite-player problem. This result is achieved as an application of a general backward propagation of chaos type result for systems of interacting forward–backward stochastic differential equations, where the interaction is heterogeneous and through the control processes, and the generator is of quadratic growth. In addition, characterising the solution of the graphon game gives rise to a novel form of infinite-dimensional forward–backward stochastic differential equation of McKean–Vlasov type, for which we provide well-posedness results. An interesting consequence of our result is the computation of the competition indifference capital, i.e., the capital making an investor indifferent between whether or not to compete.

Suggested Citation

  • Ludovic Tangpi & Xuchen Zhou, 2024. "Optimal investment in a large population of competitive and heterogeneous agents," Finance and Stochastics, Springer, vol. 28(2), pages 497-551, April.
    • Handle: RePEc:spr:finsto:v:28:y:2024:i:2:d:10.1007_s00780-023-00527-9
    DOI: 10.1007/s00780-023-00527-9
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    References listed on IDEAS

    as
    1. Guanxing Fu & Xizhi Su & Chao Zhou, 2020. "Mean Field Exponential Utility Game: A Probabilistic Approach," Papers 2006.07684, arXiv.org, revised Jul 2020.
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    3. René Carmona & Daniel B. Cooney & Christy V. Graves & Mathieu Laurière, 2022. "Stochastic Graphon Games: I. The Static Case," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 750-778, February.
    4. Goncalo dos Reis & Vadim Platonov, 2020. "Forward utility and market adjustments in relative investment-consumption games of many players," Papers 2012.01235, arXiv.org, revised Mar 2022.
    5. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    6. Frei, Christoph, 2014. "Splitting multidimensional BSDEs and finding local equilibria," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2654-2671.
    7. Freddy Delbaen & Peter Grandits & Thorsten Rheinländer & Dominick Samperi & Martin Schweizer & Christophe Stricker, 2002. "Exponential Hedging and Entropic Penalties," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 99-123, April.
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    More about this item

    Keywords

    Stochastic graphon games; Propagation of chaos; FBSDE; McKean–Vlasov equations;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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