A Stochastic Maximum Principle Of Mean-field Type With Monotonicity Conditions

Pontryagin,Boltyanski,Gamkrelidze and Mischenko[46]studied the maximum principle of deterministic systems and got the necessary conditions for optimal control.Kushner[33,34]obtained the necessary conditions for optimal control of stochastic systems with continuous parameters by using Girsanov's method.Haussmann[50]proved a more general stochastic maximum principle when the diffusion coefficient is independent of control.Later,many scholars further studied the Pontryagin's stochastic maximum principle,that is,the stochastic maximum principle under the assumption that the control domain is a convex set.Peng[44]gave a way to overcome the difficulty for the first time,and obtained a stochastic maximum principle in which the control domain is not necessarily a convex set and the diffusion term depends on the control,which is Peng's stochastic maximum principle.Buckdahn,Djehiche and Li[13]studied the stochastic control problem for a class of mean-field stochastic differential equations(SDE),and obtained Peng's stochastic maximum principle.Buckdahn,Li and Ma[15]further studied Peng's stochas-tic maximum principle when the coefficients depend on the law of state variables.Most of the works of the stochastic maximum principle,especially the mean-field stochastic maximum principle,are done under the assumptions that the coefficients satisfy Lipschitz conditions.The key point of this paper is to weaken the Lipschitz condition and study the corresponding Pontryagin's and Peng's stochastic maximum principle of mean-field type with monotonicity conditions.In this extension,we assume that the drift term satisfy monotonicity conditions not only in the state process but also in its law.In our paper,we first study the stochastic maximum principle of mean-field type under appropriate monotonicity conditions.The dynamics of our control problem is given by the following controlled stochastic differential equation:(?)(1)Our objective is to find the conditions for an optimal control u(·)which minimizes the following cost functional:(?)(2)We assume that the drift coefficient of the equation(1)satisfies the following monotonicity conditions with respect to the variables x,y:(b(t,x1,y,u)-b(t,x2,y,u)){x1-x2)??|x1-x2|2,?>0,(b(t,x,y1,u)-b(t,x,y2,u))(y1-y2)??|y1-y2|2,?>0.In order to study the stochastic maximum principle under our assumptions,we proves the existence and uniqueness of the solution to the mean-field stochastic differential equation(1)under monotonicity conditions and get some estimates,which is one of the difficulties in this paper.When the control domain is convex,by using the technique of convex perturbation and duality,we get a local form of the Pontryagin's stochastic maximum principle and give an example.On the other hand,when the control domain is not necessarily convex,we use the method proposed by Peng[44]in 1990 to study it under the assumption of the monotonicity conditions.We prove Peng's stochastic maximum principle by considering the second order variational equations and the corresponding second order adjoint processes and we give an example.In the second part of this paper,we study the stochastic maximum principle for the mean-field stochastic differential equation,in which the coefficients depend on the law of state process.We consider the following controlled stochastic differential system:(?)(3)The coefficients of the equation do not depend only on the state process but also on its law.The associated cost functional is as follows:(?)(4)u*denotes the optimal control that minimizes the cost functional.We asurume that the drift coefficient of the equation(3)satisfies the following monotonicity conditions:(b(t,x1,?,u)-b(t,x2,?,u))(x1-x2)??|x1-x2|2,?>0,(b(t,x,PY1,u)-b(t,x,PY2,u))E[Y1-Y2]? ?(W1(PY1,PY2))2,?>0.Under these conditions we prove the existence and uniqueness of the solution to the equation(3)and obtain the required estimates.Then,we prove the Pontryagin's stochastic maximum principle by using the techniques of convex perturbation and duality.Finally,when the control domain is not necessarily a convex set,we get the required estimates and prove Peng's stochastic maximum principle by using the spike variation.The difficulty of this part is that we need to consider the derivative of the corresponding functions with respect to probability measures and get relevant estimates.Different from the classical cases,the adjoint equations in this paper are mean-field backward stochastic differential equations with the monotonicity conditions,and we prove the existence and uniqueness results of their solutions.