Maximum principle for forward-backward stochastic control system under G-expectation and relation to dynamic programming

In this paper, based on the theory of stochastic differential equations on a sublinear expectation space ( ? , H , E ? ) , we develop a stochastic maximum principle for a general forward-backward stochastic control system under G -expectation. Under some convexity assumptions, we also obtain sufficient conditions for the optimality. Furthermore, relations between the adjoint processes and the value function for stochastic recursive optimal control problems are given. Finally, applications of our main results to the recursive utility portfolio optimization problem in financial market are discussed.

[1]  H. M. Soner,et al.  Martingale Representation Theorem for the G-Expectation , 2010, 1001.3802.

[2]  Shige Peng,et al.  Stopping times and related Itô's calculus with G-Brownian motion , 2009, 0910.3871.

[3]  Jörg Vorbrink.  American options with multiple priors in continuous time , 2011 .

[4]  S. Peng,et al.  Backward stochastic differential equations driven by G-Brownian motion , 2012, 1206.5889.

[5]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[6]  J. Bismut.  Conjugate convex functions in optimal stochastic control , 1973 .

[7]  S. Peng,et al.  A complete representation theorem for G-martingales , 2012, 1201.2629.

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

[9]  S. Peng,et al.  Comparison Theorem, Feynman-Kac Formula and Girsanov Transformation for BSDEs Driven by G-Brownian Motion , 2012, 1212.5403.

[10]  Jingtao Shi,et al.  Relationship between Maximum Principle and Dynamic Programming for Stochastic Recursive Optimal Control Problems and Applications , 2013 .

[11]  Optimal control with delayed information flow of systems driven by G-Brownian motion , 2014, Probability, Uncertainty and Quantitative Risk.

[12]  Shige Peng,et al.  On representation theorem of G-expectations and paths of G-Brownian motion , 2009, 0904.4519.

[13]  S. Peng.  G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty , 2007, 0711.2834.

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

[15]  Defei Zhang.  Stochastic optimal control problems under G‐expectation , 2013 .

[16]  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.

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

[18]  Zhen Wu,et al.  Maximum principle for forward-backward stochastic control system with random jumps and applications to finance , 2010, J. Syst. Sci. Complex..

[19]  Shige Peng,et al.  Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths , 2008, 0802.1240.

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

[21]  S. Peng.  Nonlinear Expectations and Stochastic Calculus under Uncertainty , 2010, Probability Theory and Stochastic Modelling.

[22]  Larry G. Epstein,et al.  Ambiguous Volatility and Asset Pricing in Continuous Time , 2012, 1301.4614.

[23]  Jingtao Shi,et al.  Relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problems of jump diffusions , 2014 .

[24]  S. Peng.  A Generalized dynamic programming principle and hamilton-jacobi-bellman equation , 1992 .

[25]  Bernt Øksendal,et al.  Maximum Principles for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps , 2009, SIAM J. Control. Optim..

[26]  Nizar Touzi,et al.  Wellposedness of second order backward SDEs , 2010, 1003.6053.

[27]  S. Peng.  G -Expectation, G -Brownian Motion and Related Stochastic Calculus of Itô Type , 2006, math/0601035.

[28]  Xiuchun Bi,et al.  Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion , 2013, 1306.0176.

[29]  S. Peng.  Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation , 2006, math/0601699.

[30]  Mark D. Schroder,et al.  Optimal Consumption and Portfolio Selection with Stochastic Differential Utility , 1999 .

[31]  Joerg Vorbrink,et al.  Financial markets with volatility uncertainty , 2010, 1012.1535.

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

[33]  S. Peng,et al.  A dynamic maximum principle for the optimization of recursive utilities under constraints , 2001 .