A mean-field maximum principle for optimal control of forward–backward stochastic differential equations with Poisson jump processes

We consider mean-field type stochastic optimal control problems for systems governed by special mean-field forward–backward stochastic differential equations with jump processes, in which the coefficients contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. Necessary conditions of optimal control for these systems in the form of a maximum principle are established by means of spike variation techniques. Our result differs from the classical one in the sense that here the adjoint equation has a mean-field type. The control domain is not assumed to be convex.

[1]  Juan Li,et al.  Stochastic maximum principle in the mean-field controls , 2012, Autom..

[2]  P. Lions,et al.  Mean field games , 2007 .

[3]  Syed Abbas,et al.  A general maximum principle for mean-field stochastic differential equations with jump processes , 2013, 1301.7327.

[4]  Ruimin Xu.  Mean-field backward doubly stochastic differential equations and related SPDEs , 2012 .

[5]  Jingtao Shi,et al.  Sufficient conditions of optimality for mean-field stochastic control problems , 2012, 2012 12th International Conference on Control Automation Robotics & Vision (ICARCV).

[6]  Abel Cadenillas.  A stochastic maximum principle for systems with jumps, with applications to finance , 2002, Syst. Control. Lett..

[7]  S. Peng,et al.  Mean-field backward stochastic differential equations and related partial differential equations , 2007, 0711.2167.

[8]  J. Yong.  A Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations , 2011, 1110.1564.

[9]  Syed Abbas,et al.  On Maximum Principle of Near-optimality for Diffusions with Jumps, with Application to Consumption-Investment Problem , 2012 .

[10]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[11]  Jingtao Shi.  Necessary Conditions for Optimal Control of Forward-Backward Stochastic Systems with Random Jumps , 2012 .

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

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

[14]  Jin Ma.  Forward-backward stochastic differential equations and their applications in finance , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[15]  Wu Zhen,et al.  Maximum Principle for Fully Coupled Forward-backward Stochastic Control System with Random Jumps , 2006, 2007 Chinese Control Conference.

[16]  Rene Carmona,et al.  Mean field forward-backward stochastic differential equations , 2012, 1211.4186.

[17]  B. Øksendal,et al.  Applied Stochastic Control of Jump Diffusions , 2004, Universitext.

[18]  Jiongmin Yong,et al.  Optimality Variational Principle for Controlled Forward-Backward Stochastic Differential Equations with Mixed Initial-Terminal Conditions , 2010, SIAM J. Control. Optim..

[19]  Zhen Wu,et al.  Maximum Principle for Partially-Observed Optimal Control of Fully-Coupled Forward-Backward Stochastic Systems , 2010 .

[20]  Boualem Djehiche,et al.  A General Stochastic Maximum Principle for SDEs of Mean-field Type , 2011 .

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

[22]  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.

[23]  Daniel Andersson,et al.  A Maximum Principle for SDEs of Mean-Field Type , 2011 .

[24]  Thomas G. Kurtz,et al.  Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type , 2008 .

[25]  N. U. Ahmed,et al.  Nonlinear Diffusion Governed by McKean--Vlasov Equation on Hilbert Space and Optimal Control , 2007, SIAM J. Control. Optim..

[26]  B. Øksendal,et al.  Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance , 2004 .

[27]  B. Bouchard,et al.  Discrete time approximation of decoupled Forward-Backward SDE with jumps , 2008 .

[28]  M. Hp.  A class of markov processes associated with nonlinear parabolic equations. , 1966 .

[29]  Shi,et al.  The Maximum Principle for Fully Coupled Forward-backward Stochastic Control System , 2006 .

[30]  Bernt Øksendal,et al.  A mean-field stochastic maximum principle via Malliavin calculus , 2012 .