The present paper considers a stochastic optimal control problem, in which the cost function is defined through a backward stochastic differential equation with infinite horizon driven by G-Brownian motion. Then we study the regularities of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the unique viscosity solution of the related Hamilton−Jacobi−Bellman−Isaacs (HJBI) equation.
Accepté le :
DOI : 10.1051/cocv/2017044
Mots-clés : G-Brownian motion, backward stochastic differential equations, stochastic optimal control, dynamic programming principle
@article{COCV_2018__24_2_873_0,
author = {Hu, Mingshang and Wang, Falei},
title = {Stochastic optimal control problem with infinite horizon driven by {G-Brownian} motion},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {873--899},
publisher = {EDP-Sciences},
volume = {24},
number = {2},
year = {2018},
doi = {10.1051/cocv/2017044},
mrnumber = {3816420},
zbl = {1401.93224},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2017044/}
}
TY - JOUR AU - Hu, Mingshang AU - Wang, Falei TI - Stochastic optimal control problem with infinite horizon driven by G-Brownian motion JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 873 EP - 899 VL - 24 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2017044/ DO - 10.1051/cocv/2017044 LA - en ID - COCV_2018__24_2_873_0 ER -
%0 Journal Article %A Hu, Mingshang %A Wang, Falei %T Stochastic optimal control problem with infinite horizon driven by G-Brownian motion %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 873-899 %V 24 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2017044/ %R 10.1051/cocv/2017044 %G en %F COCV_2018__24_2_873_0
Hu, Mingshang; Wang, Falei. Stochastic optimal control problem with infinite horizon driven by G-Brownian motion. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 873-899. doi : 10.1051/cocv/2017044. http://www.numdam.org/articles/10.1051/cocv/2017044/
[1] , and , On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57 (2008) 213–246 | DOI | MR | Zbl
[2] and , Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs. J. Funct. Anal. 155 (1998) 455–494 | DOI | MR | Zbl
[3] and , Stochastic differential games and viscosity solutions of Hamilton−Jacobi−Bellman−Isaacs equations. SIAM. J. Control. Optim. 47(1) (2008) 444–475 | DOI | MR | Zbl
[4] and , Generalized Hamilton−Jacobi−Bellman equations with Dirichlet boundary and stochastic exit time optimal control problem. SIAM. J. Control. Optim. 54 (2016) 602–631 | DOI | MR | Zbl
[5] and , Ambiguity, risk, and asset returns in continuous time. Econ. 70 (2002) 1403–1443 | MR | Zbl
[6] , and , User’s guide to viscosity solutions of second order partial differential equations. Bulletin Amer. Math. Soc. 27 (1992) 1–67 | DOI | MR | Zbl
[7] , and , Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. Potential Anal. 34 (2011) 139–161 | DOI | MR | Zbl
[8] and , A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. App. Prob. 16 (2006) 827–852 | MR | Zbl
[9] and , Optimal investment under model uncertainty in non-dominated models. SIAM. J. Control Optim. 51 (2013) 1803–1822 | DOI | MR | Zbl
[10] and , Ambiguous volatility and asset pricing in continuous time. Rev. Finan. Stud. 26 (2013) 1740–1786 | DOI
[11] and , Ambiguous volatility, possibility and utility in continuous time. J. Math. Econ. 50 (2014) 269–282 | DOI | MR | Zbl
[12] and , Control Markov Processes and Viscosity solutions. Springer, New York (1992) | MR | Zbl
[13] and , Infinite horizon BSDEs in infinite dimensions with continuous driver and applications. J. Evol. Equ. 6 (2006) 459–484 | DOI | MR | Zbl
[14] , Pathwise properties and homomorphic flows for stochastic differential equations driven by G-Brownian motion. Stoch. Proc. Appl. 119 (2009) 3356–3382 | DOI | MR | Zbl
[15] and , Dynamic programming principle for stochastic recursive optimal control problem under G-framework. Stoch. Proc. Appl. 127 (2017) 107–134 | DOI | MR | Zbl
[16] , , and , Backward stochastic differential equations driven by G-Brownian motion. Stoch. Proc. Appl. 124 (2014) 759–784 | DOI | MR | Zbl
[17] , , and , Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stoch. Proc. Appl. 124 (2014) 1170–1195 | DOI | MR | Zbl
[18] , and , A stochastic recursive optimal control problem under the G-expectation framework. Appl. Math. Optim. 70 (2014) 253–278 | DOI | MR | Zbl
[19] and , On representation theorem of G-expectations and paths of G-Brownian motion. Acta Math. Appl. Sin. Engl. Ser. 25 (2009) 539–546 | DOI | MR | Zbl
[20] and , Ergodic BSDEs driven by G-Brownian motion and their applications. Preprint (2017) | arXiv | MR
[21] , and , Quasi-continuous random variables and processes under the G-expectation framework. Stoch. Proc. Appl. 126 (2016) 2367–2387 | DOI | MR | Zbl
[22] and , On an infinite horizon and elliptic PDEs in infinite dimension. Nonlinear Differ. Equ. Appl. 14 (2007) 825–846 | DOI | MR | Zbl
[23] and , Stopping times and related Itô’s calculus with G-Brownian motion. Stoch. Proc. Appl. 121 (2011) 1492–1508 | DOI | MR | Zbl
[24] , Optimal control of diffusion processes and Hamilton−Jacobi−Bellman equations. Part 2. Commun. Partial Differ. Equ. 8 (1983) 1229–1276 | DOI | MR | Zbl
[25] and , Optimal control of stochastic integrals and Hamilton−Jacobi−Bellman equations. I. SIAM J. Control Optimiz. 20 (1982) 58–81 | DOI | MR | Zbl
[26] and , Optimal control of stochastic integrals and Hamilton−Jacobi−Bellman equations. II. SIAM J. Control Optimiz. 20 (1982) 82–95 | DOI | MR | Zbl
[27] and , Forward-backward stochastic differential equations and their applications. Lect. Notes Math., Springer 1702 (1999) | MR | Zbl
[28] , and , Robust Utility maximization in non-dominated models with 2BSDEs. Math. Finance 25 (2015) 258–287 | DOI | MR | Zbl
[29] and , Robust utility maximization with Lévy processes. Math. Finance 28 (2018) 82–105 | DOI | MR | Zbl
[30] , Comportement en temps long des équations de Hamilton−Jacobi dans des cas non standards. Ph. D. thesis, Université de Rennes 1 (2016)
[31] , Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. In Progr. Probab. edited by , , and . Birkhaüser Boston, Boston, MA 42 (1998) 79–127 | MR | Zbl
[32] , A generalized dynamic programming principle and Hamilton−Jacobi−Bellmen equation. Stochastics Stochastics Rep. 38 (1992) 119–134 | DOI | MR | Zbl
[33] , Backward stochastic differential equation and application to optimal control. Appl. Math. Optimiz. 27 (1993) 125–144 | DOI | MR | Zbl
[34] , Backward stochastic differential equations–stochastic optimization theory and viscosity solutions of HJB equations. In Topics on Stochastic Analysis, edited by , , and . Science Press, Beijing (in Chinese) (1997) 85–138
[35] , G-expectation, G-Brownian Motion and Related Stochastic Calculus of Itô type. Stochastic analysis and applications. Abel Symp. 2, Springer, Berlin (2007) 541–567 | MR | Zbl
[36] , Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch. Proc. Appl. 118 (2008) 2223–2253 | DOI | MR | Zbl
[37] , Nonlinear expectations and stochastic calculus under uncertainty. Preprint (2010) | arXiv | MR
[38] , Viscosity solutions of fully nonlinear elliptic path dependent PDEs. Ann. Appl. Probab. 26 (2016) 3381–3414 | MR | Zbl
[39] , and , An overview of viscosity solution of path dependent PDEs. Stochastic Analysis and Applications-In Honour of Terry Lyons, Springer Proceedings in Mathematics and Statistics 100 (2014) 397–453 | MR | Zbl
[40] , BSDEs with a random terminal time driven by a monotone generator and their links with PDEs. Stoch. Stoch. Rep. 76 (2004) 281–307 | DOI | MR | Zbl
[41] , and , Martingale representation theorem for the G-expectation. Stoch. Proc. Appl. 121 (2011) 265–287 | DOI | MR | Zbl
[42] , and , Wellposedness of second order backward SDEs. Prob. Theory Related Fields 153 (2012) 149–190 | DOI | MR | Zbl
[43] , Some properties on G-evaluation and its applications to G-martingale decomposition. Sci. China Math. 54 (2011) 287–300 | DOI | MR | Zbl
[44] , and , Robust utility maximization for a diffusion market model with misspecified coefficients. Finance and Stochastics 17 (2013) 535–563 | DOI | MR | Zbl
[45] and , Stochastic controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999) | DOI | MR | Zbl
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