The Differential Game Theory Of Mean-field Backward Stochastic Differential Equations With Applications

Mean-field theory has important applications in many fields, for example, financial economy, physics, chemistry, and many other fields, which attracts many scholars engaged in the related research of the mean-field theory. Lasry and Lions, they have studied the mean-field approximation problems in the field of financial and economic field, as well as N-players differential game and Nash equilibrium point, and let Nâ†'∞, then the mean-field limit equation is obtained. Buckdahn, Peng and Li studied the mean-field backward stochastic differential equations[1,7], and given the uniqueness of the solution and the comparison theorem of solution.Most of the literatures assume that the information is complete, but in fact, the decision makers can only observe the partial information, which can lead to the study of the mean-field backward stochastic system and its control theory research under partial information.Based on this, this paper studies the differential game of backward s-tochastic differential system with its applications.Firstly, we give a brief introduction of backward stochastic differential e-quation, differential games and and mean-field backward stochastic differential equation; Secondly, preliminary, we give an important theorem:the existence and uniqueness of backward stochastic differential equations, please refer to [18]; Thirdly, research on mean-field backward stochastic differential equation-s, and deduce the sufficient condition for Nash equilibrium point, as well as the necessary condition. In the study of this part,mainly using the maximum prin-ciple, the variational equations,variational inequalities for deduction, Hamil-tonian function also plays an important role. The adjoint equation is given, it provides a convenient for research. What’s moreover, we give a theorem: the existence and uniqueness of mean-field backward stochastic differential e-quations with initial coupled; Finally, we put the result which is observed in section 3 into the linear-quadratic differential game, it helps the theory to ap-plication and promotion.