January  2016, 1: 7 doi: 10.1186/s41546-016-0008-x

Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications

1 Laboratoire de Probabilités et Modèles Aléatoires, CNRS, UMR 7599, Université Paris Diderot, Paris, France;

2 CREST-ENSAE, Paris, France

Received  April 25, 2016 Revised  August 10, 2016

We consider the optimal control problem for a linear conditional McKeanVlasov equation with quadratic cost functional. The coefficients of the system and the weighting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration. Semi closed-loop strategies are introduced, and following the dynamic programming approach in (Pham and Wei, Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics, 2016), we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations. We present several financial applications with explicit solutions, and revisit, in particular, optimal tracking problems with price impact, and the conditional mean-variance portfolio selection in an incomplete market model.
Citation: Huyên Pham. Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications. Probability, Uncertainty and Quantitative Risk, doi: 10.1186/s41546-016-0008-x
References:
[1]

Alfonsi, A, Fruth, A, Schied, A:Optimal execution strategies in limit order books with general shape functions. Quantit. Finance 10, 143-157 (2010),

[2]

Almgren, R, Chriss, N:Optimal execution of portfolio transactions. J. Risk 3, 5-39 (2000),

[3]

Almgren, R, Li, TM:Market microstructure and liquidity, 2(1) (2016),

[4]

Andersson, D, Djehiche, B:A maximum principle for SDEs of mean-field type. Appl. Math. Optimization 63, 341-356 (2010),

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Bain, A, Crisan, D:Fundamentals of stochastic filtering, Series Stochastic Modelling and Applied Probability, vol. 60. Springer, New York (2009),

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Basak, S, Chabakauri, G:Dynamic mean-variance asset allocation. Rev. Finan. Stud 23, 2970-3016 (2010),

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Bensoussan, A, Frehse, J, Yam, P:Mean Field Games and Mean Field Type Control Theory. Springer(2013),

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Bismut, JM:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim 14, 419-444 (1976),

[9]

Borkar, V, Kumar, KS:McKean-Vlasov limit in portfolio optimization. Stoch. Anal. Appl. 28, 884-906(2010),

[10]

Buckdahn, R, Djehiche, B, Li, J:A general maximum principle for SDEs of mean-field type. Appl. Math.Optim 64(2), 197-216 (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:Forward-backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics. Ann. Probab 43(5), 2647-2700 (2015),

[15]

Carmona, R, Zhu, X:A probabilistic approach to mean field games with major and minor players. Ann.Appl. Prob 26(3), 1535-1580 (2016). arXiv:1409.7141v1,

[16]

Carmona, R, Delarue, F, Lachapelle, A:Control of McKean-Vlasov dynamics versus mean field games.Math. Financial Econ 7, 131-166 (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), 911-933 (2015),

[18]

Cartea, A, Jaimungal, S:A closed-form 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 School-1979. Lecture Notes Math 876, 73-238 (1981), Springer Frei, C, Westray, N:Optimal execution of a VWAP order:a stochastic control approach. Math. Finance 25, 612-639 (2015),

[21]

Hu, Y, Jin, H, Zhou, XY:Time-inconsistent stochastic linear-quadratic control. SIAM J. Control Optim 50, 1548-1572 (2012),

[22]

Huang, J, Li, X, Yong, J:A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Math. Control Related Fields 5, 97-139 (2015),

[23]

Kunita, H:Ecole d'Eté de Probabilités de Saint-Flour XII. Springer-Verlag, Berlin, New York (1982),

[24]

Li, D, Zhou, XY:Continuous-time mean-variance portfolio selection:a stochastic LQ framework. Appl.Math. Optim 42, 19-33 (2000),

[25]

Li, X, Sun, J, Yong, J:Mean-Field Stochastic Linear Quadratic Optimal Control Problems:Closed-Loop Solvability (2016). arXiv:1602.07825,

[26]

Lions, PL:Cours au Collège de France:Théorie des jeux à champ moyens, audio conference 2006-2012 (2012),

[27]

Peng, S:Stochastic Hamilton Jacobi Bellman equations. SIAM J. Control Optim 30, 284-304 (1992),

[28]

Pham, H, Wei, X:Bellman equation and viscosity solutions for mean-field stochastic control problem(2015). arXiv:1512.07866v2,

[29]

Pham, H, Wei, X:Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics(2016). arXiv:1604.04057,

[30]

Predoiu, S, Shaikhet, G, Shreve, S:Optimal execution in a general one-sided limit-order book. SIAM J.Financial Math 2, 183-212 (2011),

[31]

Rogers, LCG, Singh, S:The cost of illiquidity and its effects on hedging. Mathematical Finance 20, 597-615 (2010),

[32]

Sun, J:Mean-Field Stochastic Linear Quadratic Optimal Control Problems:Open-Loop Solvabilities(2015). arXiv:1509.02100v2,

[33]

Sun, J, Yong, J:Linear Quadratic Stochastic Differential Games:Open-Loop and Closed-Loop Saddle Points. SIAM J. Control Optim 52, 4082-4121 (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, 53-75 (2003),

[35]

Wonham, W:On a matrix Riccati equation of stochastic control. SIAM J. Control 6, 681-697 (1968),

[36]

Yong, J:A linear-quadratic optimal control problem for mean-field stochastic differential equations. SIAM J. Control Optim 51(4), 2809-2838 (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, 143-157 (2010),

[2]

Almgren, R, Chriss, N:Optimal execution of portfolio transactions. J. Risk 3, 5-39 (2000),

[3]

Almgren, R, Li, TM:Market microstructure and liquidity, 2(1) (2016),

[4]

Andersson, D, Djehiche, B:A maximum principle for SDEs of mean-field type. Appl. Math. Optimization 63, 341-356 (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 mean-variance asset allocation. Rev. Finan. Stud 23, 2970-3016 (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, 419-444 (1976),

[9]

Borkar, V, Kumar, KS:McKean-Vlasov limit in portfolio optimization. Stoch. Anal. Appl. 28, 884-906(2010),

[10]

Buckdahn, R, Djehiche, B, Li, J:A general maximum principle for SDEs of mean-field type. Appl. Math.Optim 64(2), 197-216 (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:Forward-backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics. Ann. Probab 43(5), 2647-2700 (2015),

[15]

Carmona, R, Zhu, X:A probabilistic approach to mean field games with major and minor players. Ann.Appl. Prob 26(3), 1535-1580 (2016). arXiv:1409.7141v1,

[16]

Carmona, R, Delarue, F, Lachapelle, A:Control of McKean-Vlasov dynamics versus mean field games.Math. Financial Econ 7, 131-166 (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), 911-933 (2015),

[18]

Cartea, A, Jaimungal, S:A closed-form 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 School-1979. Lecture Notes Math 876, 73-238 (1981), Springer Frei, C, Westray, N:Optimal execution of a VWAP order:a stochastic control approach. Math. Finance 25, 612-639 (2015),

[21]

Hu, Y, Jin, H, Zhou, XY:Time-inconsistent stochastic linear-quadratic control. SIAM J. Control Optim 50, 1548-1572 (2012),

[22]

Huang, J, Li, X, Yong, J:A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Math. Control Related Fields 5, 97-139 (2015),

[23]

Kunita, H:Ecole d'Eté de Probabilités de Saint-Flour XII. Springer-Verlag, Berlin, New York (1982),

[24]

Li, D, Zhou, XY:Continuous-time mean-variance portfolio selection:a stochastic LQ framework. Appl.Math. Optim 42, 19-33 (2000),

[25]

Li, X, Sun, J, Yong, J:Mean-Field Stochastic Linear Quadratic Optimal Control Problems:Closed-Loop Solvability (2016). arXiv:1602.07825,

[26]

Lions, PL:Cours au Collège de France:Théorie des jeux à champ moyens, audio conference 2006-2012 (2012),

[27]

Peng, S:Stochastic Hamilton Jacobi Bellman equations. SIAM J. Control Optim 30, 284-304 (1992),

[28]

Pham, H, Wei, X:Bellman equation and viscosity solutions for mean-field stochastic control problem(2015). arXiv:1512.07866v2,

[29]

Pham, H, Wei, X:Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics(2016). arXiv:1604.04057,

[30]

Predoiu, S, Shaikhet, G, Shreve, S:Optimal execution in a general one-sided limit-order book. SIAM J.Financial Math 2, 183-212 (2011),

[31]

Rogers, LCG, Singh, S:The cost of illiquidity and its effects on hedging. Mathematical Finance 20, 597-615 (2010),

[32]

Sun, J:Mean-Field Stochastic Linear Quadratic Optimal Control Problems:Open-Loop Solvabilities(2015). arXiv:1509.02100v2,

[33]

Sun, J, Yong, J:Linear Quadratic Stochastic Differential Games:Open-Loop and Closed-Loop Saddle Points. SIAM J. Control Optim 52, 4082-4121 (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, 53-75 (2003),

[35]

Wonham, W:On a matrix Riccati equation of stochastic control. SIAM J. Control 6, 681-697 (1968),

[36]

Yong, J:A linear-quadratic optimal control problem for mean-field stochastic differential equations. SIAM J. Control Optim 51(4), 2809-2838 (2013),

[37]

Yong, J, Zhou, XY:Stochastic controls. Hamiltonian systems and HJB equations. Springer, New York(1999),

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