Stochastic Graphon Games: II. The Linear-Quadratic Case

Alexander Aurell, René Carmona, Mathieu Laurière

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.

Original languageEnglish (US)
Article number26
JournalApplied Mathematics and Optimization
Volume85
Issue number3
DOIs
StatePublished - Jun 2022

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Applied Mathematics

Keywords

  • Continuum of players
  • Exact law of large numbers
  • Fubini extensions
  • Graphons
  • Stochastic differential games

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