In this article, we consider the problem of equilibrium price formation in an incomplete securities market consisting of one major financial firm and a large number of minor firms. They carry out continuous trading via the securities exchange to minimize their cost while facing idiosyncratic and common noises as well as stochastic order flows from their individual clients. The equilibrium price process that balances demand and supply of the securities, including the functional form of the price impact for the major firm, is derived endogenously both in the market of finite population size and in the corresponding mean field limit.
Keywords: Mean field game, Major agent, mean-field type control, controlled-FBSDEs, equilibrium price formation, market clearing
@article{COCV_2022__28_1_A21_0,
author = {Fujii, Masaaki and Takahashi, Akihiko},
title = {Equilibrium price formation with a major player and its mean field limit},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022015},
mrnumber = {4390363},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022015/}
}
TY - JOUR AU - Fujii, Masaaki AU - Takahashi, Akihiko TI - Equilibrium price formation with a major player and its mean field limit JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022015/ DO - 10.1051/cocv/2022015 LA - en ID - COCV_2022__28_1_A21_0 ER -
%0 Journal Article %A Fujii, Masaaki %A Takahashi, Akihiko %T Equilibrium price formation with a major player and its mean field limit %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022015/ %R 10.1051/cocv/2022015 %G en %F COCV_2022__28_1_A21_0
Fujii, Masaaki; Takahashi, Akihiko. Equilibrium price formation with a major player and its mean field limit. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 21. doi: 10.1051/cocv/2022015
[1] , and , An extended mean field games for storage in smart grids. J. Optim. Theory Appl. 184 (2020) 644–670. | MR | DOI
[2] , and , Mean field games models of segregation. Math. Models Methods Appl. Sci. 27 (2017) 75–113. | MR | DOI
[3] , and , Mean field games and mean field type control theory. Springer Briefs in Mathematics, NY (2013). | MR | DOI
[4] , and , Mean field games with a dominating player. Appl. Math. Optim. 74 (2016) 91–128. | MR | DOI
[5] , and , Mean field control and mean field game models with several populations. Minimax Theory Appl. 03 (2018) 173–209. | MR
[6] , Notes on Mean Field Games (2013). Available at https://www.ceremade.dauphine.fr/~cardaliaguet.
[7] , and , Remarks on Nash equilibria in mean field game models with a major player. Proc. Amer. Math. Soc. 148 (2020) 4241–4255. | MR | DOI
[8] and , Probabilistic analysis of mean-field games. SIAM J. Control. Optim. 51 (2013) 2705–34. | MR | DOI
[9] and , Forward-backward stochastic differential equations and controlled McKean-Vlasov dynamics. Ann. Probab. 43 (2015) 2647–2700. | MR | DOI
[10] and , Probabilistic Theory of Mean Field Games with Applications I. Springer International Publishing, Switzerland (2018). | MR
[11] and , Probabilistic Theory of Mean Field Games with Applications II. Springer International Publishing, Switzerland (2018). | MR
[12] , and , Mean field games with common noise. Ann. Probab. 44 (2016) 3740–803. | MR | DOI
[13] and , A probabilistic approach to mean field games with major and minor players. Ann. Appl. Probab. 26 (2016) 1535–80. | MR | DOI
[14] and , Mean-field games with differing beliefs for algorithmic trading. Math. Finance 30 (2020) 995–1034. | MR | DOI
[15] , Multi-population mean field games system with Neumann boundary conditions. J. Math. Pures Appl. 103 (2015) 1294–1315. | MR | DOI
[16] , and , Electricity price dynamics in the smart grid: a mean-field-type game perspective, 23rd International Symposium on Mathematical Theory of Networks and Systems Hong Kong University of Science and Technology, Hong Kong, July 16–20, 2018 (2018).
[17] , Mean field games of controls: on the convergence of Nash equilibria. Preprint (2020). | arXiv | MR
[18] , Extended mean field control problems: a propagation of chaos result. Preprint (2020). | arXiv | MR
[19] and , On finite population games of optimal trading. Preprint (2020). | arXiv
[20] , The derivation of ergodic mean field game equations for several population of players. Dyn. Games Appl. 3 (2013) 523–536. | MR | DOI
[21] , and , Price formation and optimal trading in intraday electricity markets. Preprint (2020). | arXiv | MR
[22] , and , Price formation and optimal trading in intraday electricity markets with a Major Player. Risks 8 (2020) 133. | DOI
[23] , and , Convex analysis for LQG systems with applications to major-minor LQG mean-field game systems. Syst. Control Lett. 142 (2020) 104734. | MR | DOI
[24] , , and , A mean field game of optimal portfolio liquidation. Math. Oper. Res. (2021) 1–32. | MR
[25] , , Mean-field leader-follower games with terminal state constraint. SIAM J. Control. Optim. 58 (2018) 2078–2113. | MR | DOI
[26] , Probabilistic approach to mean field games and mean field type control problems with multiple populations. To appear in Minimiax Theory Appl. (2019). | MR
[27] and , A mean field game approach to equilibrium pricing with market clearing condition. To appear in SIAM J. Control Optim. (2020). | MR
[28] and , Strong convergence to the mean-field limit of a finite agent equilibrium. The paper was originally titled as “A finite agent equilibrium in an incomplete market and its strong convergence to the mean-field limit”. To appear SIAM J. Financial Math. (2020). | MR
[29] , and , Economic models and mean-field games theory. Publicaoes Matematicas, IMPA, Rio, Brazil (2015). | MR
[30] , and , Regularity Theory for Mean-field game systems. Springer Briefs in Mathematics (2016). | MR | DOI
[31] and , A mean-field game approach to price formation. Dyn. Games Appl. (2020). | DOI | MR
[32] , and , Mean field games and oil production. Economica (2010).
[33] , Large-population LQG games involving a major player: the Nash certainty equivalence principle. SIAM. J. Control. Optim. 48 (2010) 3318–53. | MR | DOI
[34] , and , Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6 (2006) 221–252. | MR | DOI
[35] , and , Nash certainty equivalence in large population stochastic dynamic games: connections with the physics of interacting particle systems. Proceedings of the 45th IEEE Conference on Decision and Control (2006) 4921–4926. | DOI
[36] , and , An invariance principle in large population stochastic dynamic games. J. Syst. Sci. Complexity 20 (2007) 162–172. | MR | DOI
[37] , and , Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized -Nash equilibria. IEEE Trans. Autom. Control 52 (2007) 1560–71. | MR | DOI
[38] and , Many agent games in socio-economic systems: corruption, inspection, coalition building, network growth, security. Springer Series in Operations Research and Financial Engineering (2019). | MR
[39] , Mean field games via controlled martingales problems: Existence of Markovian equilibria. Stochastic Process. Appl. 125 (2015) 2856–94. | MR | DOI
[40] , A general characterization of the mean field limit for stochastic differential games. Probab. Theory Relat. Fields 165 (2016) 581–648. | MR | DOI
[41] and , Jeux a champ moyen I. Le cas stationnaire. C. R. Sci. Math. Acad. Paris 343 (2006) 619–625. | MR | DOI
[42] and , Jeux a champ moyen II. Horizon fini et controle optimal. C. R. Sci. Math. Acad. Paris 343 (2006) 679–684. | MR | DOI
[43] and , Mean field games. Jpn. J. Math. 2 (2007) 229–260. | MR | DOI
[44] and , Mean-field games with a major player. Comptes Rendus Math. 356 (2018) 886–890. | MR | DOI
[45] and , A mean field game of portfolio trading and its consequences on perceived correlations. Available at (2019). | arXiv
[46] and , Convergence of large population games to mean field games with interaction through the controls. Preprint (2020). | arXiv | MR
[47] and , -Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents. SIAM. J. Control. Optim. 51 (2020) 3302–31. | MR | DOI
[48] , and , A mean-field game approach to equilibrium pricing, optimal generation, and trading in solar renewable energy certificate markets. Preprint (2020). | arXiv
[49] and , Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim. 37 (1999) 825–843. | MR | DOI
[50] and , A class of globally solvable Markovian quadratic bsde systems and applications. Ann. Prob. 46 (2018) 491–550. | MR
[51] , Finding adapted solutions of forward-backward stochastic differential equations: method of continuation. Probab. Theory Related Fields 107 (1997) 537–572. | MR | DOI
[52] , Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control. Optim. 48 (2010) 4119–4156. | MR | DOI
[53] , Forward-backward stochastic differential equations with mixed initial-terminal conditions. Trans. Am. Math. Soc. 362 (2010) 1047–1096. | MR | DOI
[54] , and , Teamwise mean field competitions. (2020). | arXiv | MR
[55] and , An incomplete equilibrium with a stochastic annuity. Finance Stoch. 24 (2020) 359–382. | MR | DOI
[56] , Backward Stochastic Differential Equations. Springer, NY (2017). | MR | DOI
Cité par Sources :
All the contents expressed in this research are solely those of the author and do not represent any views or opinions of any institutions. The author is not responsible or liable in any manner for any losses and/or damages caused by the use of any contents in this research.
