# American Institute of Mathematical Sciences

January  2019, 4: 1 doi: 10.1186/s41546-018-0035-x

## Mixed deterministic and random optimal control of linear stochastic systems with quadratic costs

 1. Univ Rennes, CNRS, IRMAR-UMR 6625, 35000 Rennes, France; 2. Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433 China

Received  November 26, 2018 Revised  December 11, 2018 Published  January 2019

Fund Project: Hu acknowledges research supported by Lebesgue center of mathematics "Investissements d'avenir" program-ANR-11-LABX-0020-01, by CAESARS-ANR-15-CE05-0024 and by MFG-ANR-16-CE40-0015-01. Tang acknowledges research supported by National Science Foundation of China (Grant No. 11631004) and Science and Technology Commission of Shanghai Municipality (Grant No. 14XD1400400).

In this paper, we consider the mixed optimal control of a linear stochastic system with a quadratic cost functional, with two controllers-one can choose only deterministic time functions, called the deterministic controller, while the other can choose adapted random processes, called the random controller. The optimal control is shown to exist under suitable assumptions. The optimal control is characterized via a system of fully coupled forward-backward stochastic differential equations (FBSDEs) of mean-field type. We solve the FBSDEs via solutions of two (but decoupled) Riccati equations, and give the respective optimal feedback law for both deterministic and random controllers, using solutions of both Riccati equations. The optimal state satisfies a linear stochastic differential equation (SDE) of mean-field type. Both the singular and infinite time-horizonal cases are also addressed.
Citation: Ying Hu, Shanjian Tang. Mixed deterministic and random optimal control of linear stochastic systems with quadratic costs. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 1-. doi: 10.1186/s41546-018-0035-x
##### References:
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##### References:
 [1] Bensoussan, A.:Lectures on stochastic control. In:Mitter, S.K., Moro, A. (eds.) Nonlinear filtering and stochastic control, Proceedings of the 3rd 1981 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Held at Cortona, July 1-10, 1981 pp. 1-62. Lecture Notes in Mathematics 972. Springer-Verlag, Berlin (1982) [2] Bismut, J.M.:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 14, 419-444(1976) [3] Buckdahn, R., Li, J., Peng, S.:Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119, 3133-3154(2009) [4] Haussmann, U.G.:Optimal stationary control with state and control dependent noise. SIAM J. Control. 9, 184-198(1971) [5] Kohlmann, M., Tang, S.:Minimization of risk and linear quadratic optimal control theory. SIAM J. Control Optim. 42, 1118-1142(2003) [6] Peng, S.:Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 30, 284-304(1992) [7] 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) [8] Wonham, W.M.:On a matrix Riccati equation of stochastic control. SIAM J. Control. 6, 681-697(1968) [9] Wu, H., Zhou, X.:Stochastic frequency characteristics. SIAM J. Control Optim. 40, 557-576(2001) [10] Yong, J., Zhou, X.Y.:Stochastic Controls:Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999)
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