We verify strong rates of convergence for a time-implicit, finite-element based space-time discretization of the backward stochastic heat equation, and the forward-backward stochastic heat equation from stochastic optimal control. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise. This work is thus giving a theoretical foundation for the computational findings in Dunst and Prohl, SIAM J. Sci. Comput. 38 (2016) A2725–A2755.
Keywords: Strong error estimate with rates, backward stochastic heat equation, stochastic linear quadratic problem, forward-backward stochastic heat equation
@article{COCV_2021__27_1_A56_0,
author = {Prohl, Andreas and Wang, Yanqing},
title = {Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021052},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021052/}
}
TY - JOUR AU - Prohl, Andreas AU - Wang, Yanqing TI - Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021052/ DO - 10.1051/cocv/2021052 LA - en ID - COCV_2021__27_1_A56_0 ER -
%0 Journal Article %A Prohl, Andreas %A Wang, Yanqing %T Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021052/ %R 10.1051/cocv/2021052 %G en %F COCV_2021__27_1_A56_0
Prohl, Andreas; Wang, Yanqing. Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 54. doi: 10.1051/cocv/2021052
[1] , and , A stochastic gradient descent approach for stochastic optimal control. East Asian J. Appl. Math. 10 (2020) 635–658.
[2] , , and , An efficient numerical algorithm for solving data driven feedback control problems J. Sci. Comput. 85 (2020) Paper No. 51, 27.
[3] and , A forward scheme for backward SDEs. Stochastic Process. Appl. 117 (2007) 1793–1812.
[4] and , Time discretization and Markovian iteration for coupled FBSDEs. Ann. Appl. Probab. 18 (2008) 143–177.
[5] , Stochastic maximum principle for distributed parameter systems. J. Franklin Inst. 315 (1983) 387–406.
[6] and , Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 (2004) 175–206.
[7] , and , On the stability of the L2 projection in H1(Ω). Math. Comp. 71 (2002) 147–156.
[8] and , The mathematical theory of finite element methods. Vol. 15 of Texts in Applied Mathematics. Springer, New York, third ed. (2008).
[9] , Linear multistep schemes for BSDEs. SIAM J. Numer. Anal. 52 (2014) 2815–2836.
[10] and , The stability in L$$ and W$$1 of the L2-projection onto finite element function spaces. Math. Comp. 48 (1987) 521–532.
[11] , and , Numerical schemes for forward-backward stochastic differential equations using transposition solutions (2017) preprint.
[12] and , Partial approximate controllability for linear stochastic control systems. SIAM J. Control Optim. 57 (2019) 1209–1229.
[13] , W$$-solutions of parabolic SPDEs in general domains. Stochastic Process. Appl. 130 (2020) 1–19.
[14] and , Strong solution of backward stochastic partial differential equations in C2 domains. Probab. Theory Related Fields 154 (2012) 255–285.
[15] and , The forward-backward stochastic heat equation: numerical analysis and simulation. SIAM J. Sci. Comput. 38 (2016) A2725–A2755.
[16] , and , On multilevel Picard numerical approximations for high-dimensional nonlinearparabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. J. Sci. Comput. 79 (2019) 1534–1571.
[17] , and , Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71.
[18] , and , A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15 (2005) 2172–2202.
[19] , , , and , An efficient gradient projection method for stochastic optimal control problems. SIAM J. Numer. Anal. 55 (2017) 2982–3005.
[20] and , Error estimates for parabolic optimal control problems with control and state constraints. Comput. Optim. Appl. 56 (2013) 131–151.
[21] , , and , Optimization with PDE constraints. Vol. 23 of Mathematical Modelling: Theory and Applications. Springer, New York (2009).
[22] , and , Malliavin calculus for backward stochastic differential equations and application to numerical solutions. Ann. Appl. Probab. 21 (2011) 2379–2423.
[23] , Inverse and ill-posed problems. Vol. 55 of Inverse and Ill-posed Problems Series. Walter de Gruyter GmbH & Co. KG, Berlin (2012).
[24] , , and , Numerics for stochastic distributed parameter control systems: a finite transposition method. (2020). | arXiv
[25] and , General Pontryagin-type stochastic maximum principle and backward stochastic evolution equations in infinite dimensions. SpringerBriefs in Mathematics, Springer, Cham (2014).
[26] and , Mathematical control theory for stochastic partial differential equations. Springer (in press).
[27] , Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optim. 8 (1982) 69–95.
[28] and , The Ritz-Galerkin procedure for parabolic control problems. SIAM J. Control 11 (1973) 510–524.
[29] and , A priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints. SIAM J. Control Optim. 47 (2008) 1150–1177.
[30] , Introductory lectures on convex optimization. Vol. 87 of Applied Optimization. Kluwer Academic Publishers, Boston, MA (2004).
[31] , The Malliavin calculus and related topics. Probability and its Applications (New York), Springer-Verlag, Berlin, second ed. (2006).
[32] , Error estimates for parabolic optimal control problems with control constraints. Z. Anal. Anwendungen 23 (2004) 353–376.
[33] and , Numerical solutions of backward stochastic differential equations: a finite transposition method. C. R. Math. Acad. Sci. Paris 349 (2011) 901–903.
[34] , Transposition solutions of backward stochastic differential equations and numerical schemes, Ph.D. Thesis, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (2013).
[35] , A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Math. Control Relat. Fields 6 (2016) 489–515.
[36] , Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43 (2005) 1363–1384.
[37] , and , A unified probabilistic discretization scheme for FBSDEs: stability, consistency, and convergence analysis. SIAM J. Numer. Anal. 58 (2020) 2351–2375.
[38] and , Stochastic controls: Hamiltonian systems and HJB equations. Vol. 43 of Applications of Mathematics (New York). Springer-Verlag, New York (1999).
[39] , A numerical scheme for BSDEs. Ann. Appl. Probab. 14 (2004) 459–488.
[40] , Regularities for semilinear stochastic partial differential equations. J. Funct. Anal. 249 (2007) 454–476.
[41] , and , A new kind of accurate numerical method for backward stochastic differential equations. SIAM J. Sci. Comput. 28 (2006) 1563–1581.
Cité par Sources :
This work is supported in part by the National Natural Science Foundation of China (11801467, 11701470), and the Chongqing Natural Science Foundation (cstc2018jcyjAX0148).
