Dynamic programming principle for stochastic recursive optimal control problem with delayed systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1005-1026.

In this paper, we study one kind of stochastic recursive optimal control problem for the systems described by stochastic differential equations with delay (SDDE). In our framework, not only the dynamics of the systems but also the recursive utility depend on the past path segment of the state process in a general form. We give the dynamic programming principle for this kind of optimal control problems and show that the value function is the viscosity solution of the corresponding infinite dimensional Hamilton-Jacobi-Bellman partial differential equation.

DOI : 10.1051/cocv/2011187
Classification : 49L20, 60H10, 93E20
Mots clés : stochastic differential equation with delay, recursive optimal control problem, dynamic programming principle, Hamilton-Jacobi-Bellman equation
@article{COCV_2012__18_4_1005_0,
     author = {Chen, Li and Wu, Zhen},
     title = {Dynamic programming principle for stochastic recursive optimal control problem with delayed systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1005--1026},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {4},
     year = {2012},
     doi = {10.1051/cocv/2011187},
     mrnumber = {3019470},
     zbl = {1259.49040},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2011187/}
}
TY  - JOUR
AU  - Chen, Li
AU  - Wu, Zhen
TI  - Dynamic programming principle for stochastic recursive optimal control problem with delayed systems
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
SP  - 1005
EP  - 1026
VL  - 18
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2011187/
DO  - 10.1051/cocv/2011187
LA  - en
ID  - COCV_2012__18_4_1005_0
ER  - 
%0 Journal Article
%A Chen, Li
%A Wu, Zhen
%T Dynamic programming principle for stochastic recursive optimal control problem with delayed systems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 1005-1026
%V 18
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2011187/
%R 10.1051/cocv/2011187
%G en
%F COCV_2012__18_4_1005_0
Chen, Li; Wu, Zhen. Dynamic programming principle for stochastic recursive optimal control problem with delayed systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1005-1026. doi : 10.1051/cocv/2011187. http://archive.numdam.org/articles/10.1051/cocv/2011187/

[1] M. Chang, T. Pang and M. Pemy, Optimal control of stochastic functional differential equations with a bounded memory. Stochastic An International J. Probability & Stochastic Process 80 (2008) 69-96. | MR | Zbl

[2] D. Duffie and L.G. Epstein, Stochastic differential utility. Economicrica 60 (1992) 353-394. | MR | Zbl

[3] N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equation in finance. Math. Finance 7 (1997) 1-71. | MR | Zbl

[4] M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equation in infinite dimensional spaces : the backward stochastic differential equations approach and applications to optimal control. Ann. Probab. 30 (2002) 1397-1465. | MR | Zbl

[5] M. Fuhrman, F. Masiero and G. Tessitore, Stochastic equations with delay : optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 48 (2010) 4624-4651. | MR | Zbl

[6] B. Larssen, Dynamic programming in stochastic control of systems with delay. Stoch. Stoch. Rep. 74 (2002) 651-673. | MR | Zbl

[7] B. Larssen and N.H. Risebro, When are HJB equations for control problems with stochastic delay equations finite dimensional? Dr. Scient. thesis, University of Oslo (2003).

[8] S.E.A. Mohammed, Stochastic Functional Differential Equations, Pitman (1984). | Zbl

[9] S.E.A. Mohammed, Stochastic Differential Equations with Memory : Theory, Examples and Applications, Stochastic Analysis and Related Topics 6. The Geido Workshop (1996); Progress in Probability. Birkhauser (1998). | MR | Zbl

[10] S. Peng, A generalized dynamic programming principle and Hamilton-Jacobi-Bellmen equation. Stoch. Stoch. Rep. 38 (1992) 119-134. | MR | Zbl

[11] S. Peng, Backward stochastic differential equations-stochastic optimization theory and viscosity solution of HJB equations. Topics on Stochastic Analysis (in Chinese), edited by J. Yan, S. Peng, S. Fang and L. Wu. Science Press, Beijing (1997) 85-138.

[12] Z. Wu and Z. Yu, Dynamic programming principle for one kind of stochastic recursive optimal control problem and Hamilton-Jacobi-Bellman equation. SIAM J. Control Optim. 47 (2008) 2616-2641. | MR | Zbl

[13] J. Yong and X.Y. Zhou, Stochastic Controls. Springer-Verlag (1999). | MR | Zbl

Cité par Sources :