In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integral-differential PDEs with random field. The control problem has the control constraints o...In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integral-differential PDEs with random field. The control problem has the control constraints of obstacle type. A new gradient algorithm based on the pre-conditioner conjugate gradient algorithm （PCG） is developed for this optimal control problem. This algorithm can transform a part of the state equation matrix and co-state equation matrix into block diagonal matrix and then solve the optimal control systems iteratively. The proof of convergence for this algorithm is also discussed. Finally numerical examples of a medial size are presented to illustrate our theoretical results.展开更多
In this paper we study the one-dimensional reflected backward stochastic differential equations which are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. Usin...In this paper we study the one-dimensional reflected backward stochastic differential equations which are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. Using a penalization method, we prove the existence and uniqueness of the solutions to these equations. As an application, we show that under proper assumptions the solution of the reflected equation is the value of the related mixed optimal stopping-control problem.展开更多
基金This work was supported by National Natural Science Foundation of China （No. 11501326）.
文摘In this paper, we apply stochastic Galerkin finite element methods to the optimal control problem governed by an elliptic integral-differential PDEs with random field. The control problem has the control constraints of obstacle type. A new gradient algorithm based on the pre-conditioner conjugate gradient algorithm （PCG） is developed for this optimal control problem. This algorithm can transform a part of the state equation matrix and co-state equation matrix into block diagonal matrix and then solve the optimal control systems iteratively. The proof of convergence for this algorithm is also discussed. Finally numerical examples of a medial size are presented to illustrate our theoretical results.
基金supported by Humanities and Social Science Research Youth Fund of the Ministry of Education (11YJC790015)Economic and Financial Research Department, National Centre for Mathematics and interdisciplinary Sciences, CAS+2 种基金the Innovative Research Team Support Program of Central University of Finance and EconomicsThe second author is supported by the Mathematical Tianyuan Foundation of China(Grant No. 11126050)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20113207120002)
文摘In this paper we study the one-dimensional reflected backward stochastic differential equations which are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. Using a penalization method, we prove the existence and uniqueness of the solutions to these equations. As an application, we show that under proper assumptions the solution of the reflected equation is the value of the related mixed optimal stopping-control problem.
