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Optimal Investment in a Large Population of Competitive and Heterogeneous Agents

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  • Ludovic Tangpi
  • Xuchen Zhou

Abstract

This paper studies a stochastic utility maximization game under relative performance concerns in finite agent 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 characterizations 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, characterizing 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, 2022. "Optimal Investment in a Large Population of Competitive and Heterogeneous Agents," Papers 2202.11314, arXiv.org, revised Feb 2023.
    • Handle: RePEc:arx:papers:2202.11314
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    References listed on IDEAS

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    1. Ruimeng Hu & Thaleia Zariphopoulou, 2021. "$N$-player and Mean-field Games in It\^{o}-diffusion Markets with Competitive or Homophilous Interaction," Papers 2106.00581, arXiv.org, revised Jun 2021.
    2. Michail Anthropelos & Tianran Geng & Thaleia Zariphopoulou, 2020. "Competition in Fund Management and Forward Relative Performance Criteria," Papers 2011.00838, arXiv.org.
    3. 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.
    4. Kupper, Michael & Luo, Peng & Tangpi, Ludovic, 2019. "Multidimensional Markovian FBSDEs with super-quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 902-923.
    5. Daniel Lacker & Thaleia Zariphopoulou, 2019. "Mean field and n‐agent games for optimal investment under relative performance criteria," Mathematical Finance, Wiley Blackwell, vol. 29(4), pages 1003-1038, October.
    6. 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.
    7. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    8. Briand, Philippe & Elie, Romuald, 2013. "A simple constructive approach to quadratic BSDEs with or without delay," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2921-2939.
    9. 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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