Maximum Principle for Stochastic Recursive Optimal Control Problems Involving Impulse Controls

We consider a stochastic recursive optimal control problem in which the control variable has two components: the regular control and the impulse control. The control variable does not enter the diffusion coefficient, and the domain of the regular controls is not necessarily convex. We establish necessary optimality conditions, of the Pontryagin maximum principle type, for this stochastic optimal control problem. Sufficient optimality conditions are also given. The optimal control is obtained for an example of linear quadratic optimization problem to illustrate the applications of the theoretical results.

[1]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[2]  Wu Zhen.  MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEM OF FULLY COUPLEDFORWARD-BACKWARD STOCHASTIC SYSTEMS , 1998 .

[3]  Guangchen Wang,et al.  A Maximum Principle for Partial Information Backward Stochastic Control Problems with Applications , 2009, SIAM J. Control. Optim..

[4]  Wensheng Xu,et al.  Stochastic maximum principle for optimal control problem of forward and backward system , 1995, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[5]  S. Peng.  A general stochastic maximum principle for optimal control problems , 1990 .

[6]  M. Jeanblanc-Picqué,et al.  Impulse Control Method and Exchange Rate , 1993 .

[7]  A. R. Norman,et al.  Portfolio Selection with Transaction Costs , 1990, Math. Oper. Res..

[8]  U. Haussmann,et al.  The stochastic maximum principle for a singular control problem , 1994 .

[9]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[10]  A. Chala,et al.  The stochastic maximum principle in optimal control of singular diffusions with non linear coefficients , 2005 .

[11]  Ralf Korn,et al.  Some applications of impulse control in mathematical finance , 1999, Math. Methods Oper. Res..

[12]  Zhen Wu,et al.  A maximum principle for partially observed optimal control of forward-backward stochastic control systems , 2010, Science China Information Sciences.

[13]  S. Peng,et al.  Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control , 1999 .

[14]  Xunjing Li,et al.  Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps , 1994 .

[15]  Zhiyong Yu,et al.  A partial information non-zero sum differential game of backward stochastic differential equations with applications , 2012, Autom..

[16]  Zhiyong Yu,et al.  A Pontryagin's Maximum Principle for Non-Zero Sum Differential Games of BSDEs with Applications , 2010, IEEE Transactions on Automatic Control.

[17]  Zhen Wu,et al.  The Maximum Principles for Stochastic Recursive Optimal Control Problems Under Partial Information , 2009, IEEE Transactions on Automatic Control.

[18]  Larry G. Epstein,et al.  Stochastic differential utility , 1992 .

[19]  B. Mezerdi,et al.  A General Stochastic Maximum Principle for Singular Control Problems , 2005 .

[20]  Zhen Wu,et al.  Stochastic Maximum Principle for Optimal Control Problems of Forward-Backward Systems Involving Impulse Controls , 2011, IEEE Transactions on Automatic Control.

[21]  Boris M. Miller,et al.  Maximum Principle for Singular Stochastic Control Problems , 2006, SIAM J. Control. Optim..

[22]  Hai-ping Shi.  Backward stochastic differential equations in finance , 2010 .