This paper is concerned with a linear-quadratic (LQ, for short) optimal control problem for backward stochastic differential equations (BSDEs, for short), where the coefficients of the backward control system and the weighting matrices in the cost functional are allowed to be random. By a variational method, the optimality system, which is a coupled linear forward-backward stochastic differential equation (FBSDE, for short), is derived, and by a Hilbert space method, the unique solvability of the optimality system is obtained. In order to construct the optimal control, a new stochastic Riccati-type equation is introduced. It is proved that an adapted solution (possibly non-unique) to the Riccati equation exists and decouples the optimality system. With this solution, the optimal control is obtained in an explicit way.
Keywords: Linear-quadratic optimal control, backward stochastic differential equation, random coefficient, stochastic Riccati equation
@article{COCV_2021__27_1_A48_0,
author = {Sun, Jingrui and Wang, Hanxiao},
title = {Linear-quadratic optimal control for backward stochastic differential equations with random coefficients},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021049},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021049/}
}
TY - JOUR AU - Sun, Jingrui AU - Wang, Hanxiao TI - Linear-quadratic optimal control for backward stochastic differential equations with random coefficients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021049/ DO - 10.1051/cocv/2021049 LA - en ID - COCV_2021__27_1_A48_0 ER -
%0 Journal Article %A Sun, Jingrui %A Wang, Hanxiao %T Linear-quadratic optimal control for backward stochastic differential equations with random coefficients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021049/ %R 10.1051/cocv/2021049 %G en %F COCV_2021__27_1_A48_0
Sun, Jingrui; Wang, Hanxiao. Linear-quadratic optimal control for backward stochastic differential equations with random coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 46. doi: 10.1051/cocv/2021049
[1] , and , Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM J. Control Optim. 40 (2001) 1296–1311.
[2] , and , Optimal control for controllable stochastic linear systems. ESAIM: COCV 26 (2020) 98.
[3] , Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 14 (1976) 419–444.
[4] , and , Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Control Optim. 36 (1998) 1685–1702.
[5] and , Stochastic linear quadratic optimal control problems. Appl. Math. Optim. 43 (2001) 21–45.
[6] and , Stochastic linear quadratic regulators with indefinite control weight costs. II. SIAM J. Control Optim. 39 (2000) 1065–1081.
[7] , and , Linear quadratic mean-field-game of backward stochastic differential systems. Math. Control Relat. Fields 8 (2018) 653–678.
[8] , and , Backward mean-field linear-quadratic-Gaussian (LQG) games: full and partial information. IEEE Trans. Automat. Control 61 (2016) 3784–3796.
[9] and , Multidimensional backward stochastic Riccati equations and applications. SIAM J. Control Optim. 41 (2003) 1696–1721.
[10] , and , Linear quadratic optimal control problems for mean-field backward stochastic differential equations. Appl. Math. Optim. 80 (2019) 223–250.
[11] , Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market. Math. Oper. Res. 29 (2004) 132–161.
[12] and , Linear-quadratic control of backward stochastic differential equations. SIAM J. Control Optim. 40 (2001) 450–474.
[13] and , Forward-Backward Stochastic Differential Equations and Their Applications. Vol. 1702 of Lecture Notes Math. Springer-Verlag, Berlin (1999).
[14] , Open problems on backward stochastic differential equations, in Control of Distributed Parameter and Stochastic Systems. edited by , , and . Springer, Boston, MA (1999) 265–273.
[15] , and , Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71.
[16] , and , Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems. SIAM J. Control Optim. 54 (2016) 2274–2308.
[17] , and , Stochastic linear-quadratic optimal control problems with random coefficients: Closed-loop representation of open-loop optimal controls. Ann. Appl. Probab. 31 (2021) 460–499.
[18] and , Stochastic Linear-Quadratic Optimal Control Theory: Open-Loop and Closed-Loop Solutions. Springer Briefs in Mathematics (2020).
[19] , General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim. 42 (2003) 53–75.
[20] , Dynamic programming for general linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim. 53 (2015) 1082–1106.
[21] , and , Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discete Contin. Dyn. Syst. 39 (2019) 2785–2805.
[22] , and , A kind of LQ non-zero sum differential game of backward stochastic differential equation with asymmetric information. Automatica 97 (2018) 346–352.
[23] , On a matrix Riccati equation of stochastic control. SIAM J. Control 6 (1968) 681–697.
[24] , A leader-follower stochastic linear quadratic differential game. SIAM J. Control Optim. 41 (2002) 1015–1041.
[25] and , Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999).
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