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<p>Mean Field Games (MFG) theory describes strategic interactions in differential games with a large number of small and indistinguishable players. Traditionally, the players’ control impacts only the drift term in the system’s dynamics, leaving the diffusion term uncontrolled. This paper explores a novel scenario where agents control both drift and diffusion. This leads to a fully non-linear MFG system with a fully non-linear Hamilton–Jacobi–Bellman equation. We use viscosity arguments to prove existence of solutions for the HJB equation, and then we adapt and extend a result from Krylov to prove a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_323_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {C}}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> regularity for <i>u</i> in the space variable. This allows us to prove a well-posedness result for the MFG system.</p>

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A second-order Mean Field Games model with controlled diffusion

  • 期刊论文

  • 发表日期：

    2025年4月3日

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  • Vincenzo Ignazio,
  • Michele Ricciardi

摘要

Mean Field Games (MFG) theory describes strategic interactions in differential games with a large number of small and indistinguishable players. Traditionally, the players’ control impacts only the drift term in the system’s dynamics, leaving the diffusion term uncontrolled. This paper explores a novel scenario where agents control both drift and diffusion. This leads to a fully non-linear MFG system with a fully non-linear Hamilton–Jacobi–Bellman equation. We use viscosity arguments to prove existence of solutions for the HJB equation, and then we adapt and extend a result from Krylov to prove a \({\mathcal {C}}^3\) C 3 regularity for u in the space variable. This allows us to prove a well-posedness result for the MFG system.