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Backwardforward linearquadratic meanfield games with major and minor agents
Linear quadratic optimal control of conditional McKeanVlasov equation with random coefficients and applications
1 Laboratoire de Probabilités et Modèles Aléatoires, CNRS, UMR 7599, Université Paris Diderot, Paris, France; 
2 CRESTENSAE, Paris, France 
References:
[1] 
Alfonsi, A, Fruth, A, Schied, A:Optimal execution strategies in limit order books with general shape functions. Quantit. Finance 10, 143157 (2010), 
[2] 
Almgren, R, Chriss, N:Optimal execution of portfolio transactions. J. Risk 3, 539 (2000), 
[3] 
Almgren, R, Li, TM:Market microstructure and liquidity, 2(1) (2016), 
[4] 
Andersson, D, Djehiche, B:A maximum principle for SDEs of meanfield type. Appl. Math. Optimization 63, 341356 (2010), 
[5] 
Bain, A, Crisan, D:Fundamentals of stochastic filtering, Series Stochastic Modelling and Applied Probability, vol. 60. Springer, New York (2009), 
[6] 
Basak, S, Chabakauri, G:Dynamic meanvariance asset allocation. Rev. Finan. Stud 23, 29703016 (2010), 
[7] 
Bensoussan, A, Frehse, J, Yam, P:Mean Field Games and Mean Field Type Control Theory. Springer(2013), 
[8] 
Bismut, JM:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim 14, 419444 (1976), 
[9] 
Borkar, V, Kumar, KS:McKeanVlasov limit in portfolio optimization. Stoch. Anal. Appl. 28, 884906(2010), 
[10] 
Buckdahn, R, Djehiche, B, Li, J:A general maximum principle for SDEs of meanfield type. Appl. Math.Optim 64(2), 197216 (2011), 
[11] 
Cai, J, Rosenbaum, M, Tankov, P:Asymptotic lower bounds for optimal tracking:a linear programming approach (2015). arXiv:1510.04295, 
[12] 
Cardaliaguet, P:Notes on mean field games. Notes from P.L. Lions lectures at Collège de France (2012). https://www.ceremade.dauphine.fr/cardalia/MFG100629.pdf, 
[13] 
Carmona, R, Delarue, F:The Master equation for large population equilibriums. In:Crisan, D, et al. (eds.)Stochastic Analysis and Applications 2014, Springer Proceedings in Mathematics and Statistics 100.Springer (2014), 
[14] 
Carmona, R, Delarue, F:Forwardbackward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics. Ann. Probab 43(5), 26472700 (2015), 
[15] 
Carmona, R, Zhu, X:A probabilistic approach to mean field games with major and minor players. Ann.Appl. Prob 26(3), 15351580 (2016). arXiv:1409.7141v1, 
[16] 
Carmona, R, Delarue, F, Lachapelle, A:Control of McKeanVlasov dynamics versus mean field games.Math. Financial Econ 7, 131166 (2013), 
[17] 
Carmona, R, Fouque, JP, Sun, LH:Mean field games and systemic risk. to appear in Communications in Mathematical Sciences. Communications in Mathematical Sciences 13(4), 911933 (2015), 
[18] 
Cartea, A, Jaimungal, S:A closedform execution strategy to target VWAP. to appear in SIAM Journal of Financial Mathematics (2015). Preprint available at https://papers.ssrn.com/sol3/papers.cfm?abstractid=2542314, 
[19] 
Chassagneux, JF, Crisan, D, Delarue, F:A probabilistic approach to classical solutions of the master equation for large population equilibria (2015). arXiv:1411.3009, 
[20] 
El Karoui, N:Les aspects probabilistes du contrôle stochastique. Ninth Saint Flour Probability Summer School1979. Lecture Notes Math 876, 73238 (1981), Springer Frei, C, Westray, N:Optimal execution of a VWAP order:a stochastic control approach. Math. Finance 25, 612639 (2015), 
[21] 
Hu, Y, Jin, H, Zhou, XY:Timeinconsistent stochastic linearquadratic control. SIAM J. Control Optim 50, 15481572 (2012), 
[22] 
Huang, J, Li, X, Yong, J:A linearquadratic optimal control problem for meanfield stochastic differential equations in infinite horizon. Math. Control Related Fields 5, 97139 (2015), 
[23] 
Kunita, H:Ecole d'Eté de Probabilités de SaintFlour XII. SpringerVerlag, Berlin, New York (1982), 
[24] 
Li, D, Zhou, XY:Continuoustime meanvariance portfolio selection:a stochastic LQ framework. Appl.Math. Optim 42, 1933 (2000), 
[25] 
Li, X, Sun, J, Yong, J:MeanField Stochastic Linear Quadratic Optimal Control Problems:ClosedLoop Solvability (2016). arXiv:1602.07825, 
[26] 
Lions, PL:Cours au Collège de France:Théorie des jeux à champ moyens, audio conference 20062012 (2012), 
[27] 
Peng, S:Stochastic Hamilton Jacobi Bellman equations. SIAM J. Control Optim 30, 284304 (1992), 
[28] 
Pham, H, Wei, X:Bellman equation and viscosity solutions for meanfield stochastic control problem(2015). arXiv:1512.07866v2, 
[29] 
Pham, H, Wei, X:Dynamic programming for optimal control of stochastic McKeanVlasov dynamics(2016). arXiv:1604.04057, 
[30] 
Predoiu, S, Shaikhet, G, Shreve, S:Optimal execution in a general onesided limitorder book. SIAM J.Financial Math 2, 183212 (2011), 
[31] 
Rogers, LCG, Singh, S:The cost of illiquidity and its effects on hedging. Mathematical Finance 20, 597615 (2010), 
[32] 
Sun, J:MeanField Stochastic Linear Quadratic Optimal Control Problems:OpenLoop Solvabilities(2015). arXiv:1509.02100v2, 
[33] 
Sun, J, Yong, J:Linear Quadratic Stochastic Differential Games:OpenLoop and ClosedLoop Saddle Points. SIAM J. Control Optim 52, 40824121 (2014), 
[34] 
Tang, S:General linear quadratic optimal stochastic control problems with random coefficients:linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim 42, 5375 (2003), 
[35] 
Wonham, W:On a matrix Riccati equation of stochastic control. SIAM J. Control 6, 681697 (1968), 
[36] 
Yong, J:A linearquadratic optimal control problem for meanfield stochastic differential equations. SIAM J. Control Optim 51(4), 28092838 (2013), 
[37] 
Yong, J, Zhou, XY:Stochastic controls. Hamiltonian systems and HJB equations. Springer, New York(1999), 
show all references
References:
[1] 
Alfonsi, A, Fruth, A, Schied, A:Optimal execution strategies in limit order books with general shape functions. Quantit. Finance 10, 143157 (2010), 
[2] 
Almgren, R, Chriss, N:Optimal execution of portfolio transactions. J. Risk 3, 539 (2000), 
[3] 
Almgren, R, Li, TM:Market microstructure and liquidity, 2(1) (2016), 
[4] 
Andersson, D, Djehiche, B:A maximum principle for SDEs of meanfield type. Appl. Math. Optimization 63, 341356 (2010), 
[5] 
Bain, A, Crisan, D:Fundamentals of stochastic filtering, Series Stochastic Modelling and Applied Probability, vol. 60. Springer, New York (2009), 
[6] 
Basak, S, Chabakauri, G:Dynamic meanvariance asset allocation. Rev. Finan. Stud 23, 29703016 (2010), 
[7] 
Bensoussan, A, Frehse, J, Yam, P:Mean Field Games and Mean Field Type Control Theory. Springer(2013), 
[8] 
Bismut, JM:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim 14, 419444 (1976), 
[9] 
Borkar, V, Kumar, KS:McKeanVlasov limit in portfolio optimization. Stoch. Anal. Appl. 28, 884906(2010), 
[10] 
Buckdahn, R, Djehiche, B, Li, J:A general maximum principle for SDEs of meanfield type. Appl. Math.Optim 64(2), 197216 (2011), 
[11] 
Cai, J, Rosenbaum, M, Tankov, P:Asymptotic lower bounds for optimal tracking:a linear programming approach (2015). arXiv:1510.04295, 
[12] 
Cardaliaguet, P:Notes on mean field games. Notes from P.L. Lions lectures at Collège de France (2012). https://www.ceremade.dauphine.fr/cardalia/MFG100629.pdf, 
[13] 
Carmona, R, Delarue, F:The Master equation for large population equilibriums. In:Crisan, D, et al. (eds.)Stochastic Analysis and Applications 2014, Springer Proceedings in Mathematics and Statistics 100.Springer (2014), 
[14] 
Carmona, R, Delarue, F:Forwardbackward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics. Ann. Probab 43(5), 26472700 (2015), 
[15] 
Carmona, R, Zhu, X:A probabilistic approach to mean field games with major and minor players. Ann.Appl. Prob 26(3), 15351580 (2016). arXiv:1409.7141v1, 
[16] 
Carmona, R, Delarue, F, Lachapelle, A:Control of McKeanVlasov dynamics versus mean field games.Math. Financial Econ 7, 131166 (2013), 
[17] 
Carmona, R, Fouque, JP, Sun, LH:Mean field games and systemic risk. to appear in Communications in Mathematical Sciences. Communications in Mathematical Sciences 13(4), 911933 (2015), 
[18] 
Cartea, A, Jaimungal, S:A closedform execution strategy to target VWAP. to appear in SIAM Journal of Financial Mathematics (2015). Preprint available at https://papers.ssrn.com/sol3/papers.cfm?abstractid=2542314, 
[19] 
Chassagneux, JF, Crisan, D, Delarue, F:A probabilistic approach to classical solutions of the master equation for large population equilibria (2015). arXiv:1411.3009, 
[20] 
El Karoui, N:Les aspects probabilistes du contrôle stochastique. Ninth Saint Flour Probability Summer School1979. Lecture Notes Math 876, 73238 (1981), Springer Frei, C, Westray, N:Optimal execution of a VWAP order:a stochastic control approach. Math. Finance 25, 612639 (2015), 
[21] 
Hu, Y, Jin, H, Zhou, XY:Timeinconsistent stochastic linearquadratic control. SIAM J. Control Optim 50, 15481572 (2012), 
[22] 
Huang, J, Li, X, Yong, J:A linearquadratic optimal control problem for meanfield stochastic differential equations in infinite horizon. Math. Control Related Fields 5, 97139 (2015), 
[23] 
Kunita, H:Ecole d'Eté de Probabilités de SaintFlour XII. SpringerVerlag, Berlin, New York (1982), 
[24] 
Li, D, Zhou, XY:Continuoustime meanvariance portfolio selection:a stochastic LQ framework. Appl.Math. Optim 42, 1933 (2000), 
[25] 
Li, X, Sun, J, Yong, J:MeanField Stochastic Linear Quadratic Optimal Control Problems:ClosedLoop Solvability (2016). arXiv:1602.07825, 
[26] 
Lions, PL:Cours au Collège de France:Théorie des jeux à champ moyens, audio conference 20062012 (2012), 
[27] 
Peng, S:Stochastic Hamilton Jacobi Bellman equations. SIAM J. Control Optim 30, 284304 (1992), 
[28] 
Pham, H, Wei, X:Bellman equation and viscosity solutions for meanfield stochastic control problem(2015). arXiv:1512.07866v2, 
[29] 
Pham, H, Wei, X:Dynamic programming for optimal control of stochastic McKeanVlasov dynamics(2016). arXiv:1604.04057, 
[30] 
Predoiu, S, Shaikhet, G, Shreve, S:Optimal execution in a general onesided limitorder book. SIAM J.Financial Math 2, 183212 (2011), 
[31] 
Rogers, LCG, Singh, S:The cost of illiquidity and its effects on hedging. Mathematical Finance 20, 597615 (2010), 
[32] 
Sun, J:MeanField Stochastic Linear Quadratic Optimal Control Problems:OpenLoop Solvabilities(2015). arXiv:1509.02100v2, 
[33] 
Sun, J, Yong, J:Linear Quadratic Stochastic Differential Games:OpenLoop and ClosedLoop Saddle Points. SIAM J. Control Optim 52, 40824121 (2014), 
[34] 
Tang, S:General linear quadratic optimal stochastic control problems with random coefficients:linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim 42, 5375 (2003), 
[35] 
Wonham, W:On a matrix Riccati equation of stochastic control. SIAM J. Control 6, 681697 (1968), 
[36] 
Yong, J:A linearquadratic optimal control problem for meanfield stochastic differential equations. SIAM J. Control Optim 51(4), 28092838 (2013), 
[37] 
Yong, J, Zhou, XY:Stochastic controls. Hamiltonian systems and HJB equations. Springer, New York(1999), 
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